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CONVERSION FACTORS FROM ENGLISH TO SI UNITS Length: Area: Volume: Force: 1 ft 1 ft 1 ft 1 in. 1 in. 1 in. 2 5 5 5 5 5 5 0.3048 m 30.48 cm 304.8 mm 0.0254 m 2.54 cm 25.4 mm 24 Stress: 1 lb>ft 2 1 lb>ft 2 1 U.S. ton>ft 2 1 kip>ft 2 1 lb>in2 5 5 5 5 5 Unit weight: 1 lb>ft 3 1 lb>in3 5 0.1572 kN>m3 5 271.43 kN>m3 Moment: 1 lb-ft 1 lb-in. 5 1.3558 N # m 5 0.11298 N # m Energy: 1 ft-lb 5 1.3558 J Moment of inertia: 1 in4 1 in4 5 0.4162 3 106 mm4 5 0.4162 3 1026 m4 Section modulus: 1 in3 1 in3 5 0.16387 3 105 mm3 5 0.16387 3 1024 m3 Hydraulic conductivity: 1 ft>min 1 ft>min 1 ft>min 1 ft>sec 1 ft>sec 1 in.>min 1 in.>sec 1 in.>sec 5 0.3048 m>min 5 30.48 cm>min 5 304.8 mm>min 5 0.3048 m>sec 5 304.8 mm>sec 5 0.0254 m>min 5 2.54 cm>sec 5 25.4 mm>sec 2 1 ft 1 ft 2 1 ft 2 1 in2 1 in2 1 in2 5 5 5 5 5 5 929.03 3 10 m 929.03 cm2 929.03 3 102 mm2 6.452 3 1024 m2 6.452 cm2 645.16 mm2 1 ft 3 1 ft 3 1 in3 1 in3 5 28.317 3 1023 m3 5 28.317 3 103 cm3 5 16.387 3 1026 m3 5 16.387 cm3 1 lb 1 lb 1 lb 1 kip 1 U.S. ton 1 lb 1 lb>ft 5 4.448 N 5 4.448 3 1023 kN 5 0.4536 kgf 5 4.448 kN 5 8.896 kN 5 0.4536 3 1023 metric ton 5 14.593 N>m Coefficient of consolidation: 1 in2>sec 1 in2>sec 1 ft 2>sec 47.88 N>m2 0.04788 kN>m2 95.76 kN>m2 47.88 kN>m2 6.895 kN>m2 5 6.452 cm2>sec 5 20.346 3 103 m2>yr 5 929.03 cm2>sec Principles of Foundation Engineering, SI Seventh Edition BRAJA M. DAS Australia • Brazil • Japan • Korea • Mexico • Singapore • Spain • United Kingdom • United States Principles of Foundation Engineering, SI Seventh Edition Author Braja M. Das Publisher, Global Engineering: Christopher M. Shortt Senior Developmental Editor: Hilda Gowans Editorial Assistant: Tanya Altieri Team Assistant: Carly Rizzo Marketing Manager: Lauren Betsos Production Manager: Patricia M. Boies Content Project Manager: Darrell Frye Production Service: RPK Editorial Services, Inc. Copyeditor: Shelly Gerger-Knecthl Proofreader: Martha McMaster Indexer: Braja M. Das ©2011, 2007 Cengage Learning ALL RIGHTS RESERVED. 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For your course and learning solutions, visit www.cengage.com/engineering Purchase any of our products at your local college store or at our preferred online store www.CengageBrain.com Printed in the United States of America 1 2 3 4 5 6 7 13 12 11 10 09 To our granddaughter, Elizabeth Madison This page intentionally left blank This page intentionally left blank This page intentionally left blank Contents Preface xvii 1 Geotechnical Properties of Soil 1 1.1 Introduction 1 1.2 Grain-Size Distribution 2 1.3 Size Limits for Soils 5 1.4 Weight–Volume Relationships 5 1.5 Relative Density 10 1.6 Atterberg Limits 15 1.7 Liquidity Index 16 1.8 Activity 17 1.9 Soil Classification Systems 17 1.10 Hydraulic Conductivity of Soil 25 1.11 Steady-State Seepage 28 1.12 Effective Stress 30 1.13 Consolidation 32 1.14 Calculation of Primary Consolidation Settlement 37 1.15 Time Rate of Consolidation 38 1.16 Degree of Consolidation Under Ramp Loading 44 1.17 Shear Strength 47 1.18 Unconfined Compression Test 52 1.19 Comments on Friction Angle, fr 54 1.20 Correlations for Undrained Shear Strength, Cu 57 1.21 Sensitivity 57 Problems 58 References 62 vii viii Contents 2 Natural Soil Deposits and Subsoil Exploration 64 2.1 Introduction 64 Natural Soil Deposits 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 64 Soil Origin 64 Residual Soil 66 Gravity Transported Soil 67 Alluvial Deposits 68 Lacustrine Deposits 70 Glacial Deposits 70 Aeolian Soil Deposits 71 Organic Soil 73 Some Local Terms for Soils 73 Subsurface Exploration 74 2.11 Purpose of Subsurface Exploration 74 2.12 Subsurface Exploration Program 74 2.13 Exploratory Borings in the Field 77 2.14 Procedures for Sampling Soil 81 2.15 Split-Spoon Sampling 81 2.16 Sampling with a Scraper Bucket 89 2.17 Sampling with a Thin-Walled Tube 90 2.18 Sampling with a Piston Sampler 92 2.19 Observation of Water Tables 92 2.20 Vane Shear Test 94 2.21 Cone Penetration Test 98 2.22 Pressuremeter Test (PMT) 107 2.23 Dilatometer Test 110 2.24 Coring of Rocks 113 2.25 Preparation of Boring Logs 117 2.26 Geophysical Exploration 118 2.27 Subsoil Exploration Report 126 Problems 126 References 130 3 Shallow Foundations: Ultimate Bearing Capacity 133 3.1 3.2 3.3 3.4 Introduction 133 General Concept 133 Terzaghi’s Bearing Capacity Theory 136 Factor of Safety 140 Contents 3.5 3.6 3.7 3.8 3.9 3.10 Modification of Bearing Capacity Equations for Water Table 142 The General Bearing Capacity Equation 143 Case Studies on Ultimate Bearing Capacity 148 Effect of Soil Compressibility 153 Eccentrically Loaded Foundations 157 Ultimate Bearing Capacity under Eccentric Loading—One-Way Eccentricity 159 3.11 Bearing Capacity—Two-way Eccentricity 165 3.12 Bearing Capacity of a Continuous Foundation Subjected to Eccentric Inclined Loading 173 Problems 177 References 179 4 Ultimate Bearing Capacity of Shallow Foundations: Special Cases 181 4.1 Introduction 181 4.2 Foundation Supported by a Soil with a Rigid Base at Shallow Depth 181 4.3 Bearing Capacity of Layered Soils: Stronger Soil Underlain by Weaker Soil 190 4.4 Bearing Capacity of Layered Soil: Weaker Soil Underlain by Stronger Soil 198 4.5 Closely Spaced Foundations—Effect on Ultimate Bearing Capacity 200 4.6 Bearing Capacity of Foundations on Top of a Slope 203 4.7 Seismic Bearing Capacity of a Foundation at the Edge of a Granular Soil Slope 209 4.8 Bearing Capacity of Foundations on a Slope 210 4.9 Foundations on Rock 212 4.10 Uplift Capacity of Foundations 213 Problems 219 References 221 5 Shallow Foundations: Allowable Bearing Capacity and Settlement 223 5.1 Introduction 223 Vertical Stress Increase in a Soil Mass Caused by Foundation Load 5.2 Stress Due to a Concentrated Load 224 224 ix x Contents 5.3 Stress Due to a Circularly Loaded Area 224 5.4 Stress below a Rectangular Area 226 5.5 Average Vertical Stress Increase Due to a Rectangularly Loaded Area 232 5.6 Stress Increase under an Embankment 236 5.7 Westergaard’s Solution for Vertical Stress Due to a Point Load 240 5.8 Stress Distribution for Westergaard Material 241 Elastic Settlement 243 Elastic Settlement of Foundations on Saturated Clay (S ⫽ 0.5) 243 Settlement Based on the Theory of Elasticity 245 Improved Equation for Elastic Settlement 254 Settlement of Sandy Soil: Use of Strain Influence Factor 258 Settlement of Foundation on Sand Based on Standard Penetration Resistance 263 5.14 Settlement in Granular Soil Based on Pressuremeter Test (PMT) 267 5.9 5.10 5.11 5.12 5.13 Consolidation Settlement 273 5.15 Primary Consolidation Settlement Relationships 273 5.16 Three-Dimensional Effect on Primary Consolidation Settlement 274 5.17 Settlement Due to Secondary Consolidation 278 5.18 Field Load Test 280 5.19 Presumptive Bearing Capacity 282 5.20 Tolerable Settlement of Buildings 283 Problems 285 References 288 6 Mat Foundations 291 6.1 Introduction 291 6.2 Combined Footings 291 6.3 Common Types of Mat Foundations 294 6.4 Bearing Capacity of Mat Foundations 296 6.5 Differential Settlement of Mats 299 6.6 Field Settlement Observations for Mat Foundations 300 6.7 Compensated Foundation 300 6.8 Structural Design of Mat Foundations 304 Problems 322 References 323 Contents 7 Lateral Earth Pressure 324 7.1 Introduction 324 7.2 Lateral Earth Pressure at Rest 325 Active Pressure 7.3 7.4 7.5 7.6 7.7 7.8 7.9 328 Rankine Active Earth Pressure 328 A Generalized Case for Rankine Active Pressure 334 Coulomb’s Active Earth Pressure 340 Active Earth Pressure Due to Surcharge 348 Active Earth Pressure for Earthquake Conditions 350 Active Pressure for Wall Rotation about the Top: Braced Cut 355 Active Earth Pressure for Translation of Retaining Wall—Granular Backfill 357 Passive Pressure 360 7.10 Rankine Passive Earth Pressure 360 7.11 Rankine Passive Earth Pressure: Vertical Backface and Inclined Backfill 363 7.12 Coulomb’s Passive Earth Pressure 365 7.13 Comments on the Failure Surface Assumption for Coulomb’s Pressure Calculations 366 7.14 Passive Pressure under Earthquake Conditions 370 Problems 371 References 373 8 Retaining Walls 375 8.1 Introduction 375 Gravity and Cantilever Walls 8.2 8.3 8.4 8.5 8.6 8.7 8.8 8.9 8.10 377 Proportioning Retaining Walls 377 Application of Lateral Earth Pressure Theories to Design 378 Stability of Retaining Walls 380 Check for Overturning 382 Check for Sliding along the Base 384 Check for Bearing Capacity Failure 387 Construction Joints and Drainage from Backfill 396 Gravity Retaining-Wall Design for Earthquake Conditions 399 Comments on Design of Retaining Walls and a Case Study 402 Mechanically Stabilized Retaining Walls 8.11 Soil Reinforcement 405 405 xi xii Contents 8.12 8.13 8.14 8.15 Considerations in Soil Reinforcement 406 General Design Considerations 409 Retaining Walls with Metallic Strip Reinforcement 410 Step-by-Step-Design Procedure Using Metallic Strip Reinforcement 417 8.16 Retaining Walls with Geotextile Reinforcement 422 8.17 Retaining Walls with Geogrid Reinforcement—General 428 8.18 Design Procedure fore Geogrid-Reinforced Retaining Wall 428 Problems 433 References 435 9 Sheet Pile Walls 437 9.1 9.2 9.3 9.4 9.5 Introduction 437 Construction Methods 441 Cantilever Sheet Pile Walls 442 Cantilever Sheet Piling Penetrating Sandy Soils 442 Special Cases for Cantilever Walls Penetrating a Sandy Soil 449 9.6 Cantilever Sheet Piling Penetrating Clay 452 9.7 Special Cases for Cantilever Walls Penetrating Clay 457 9.8 Anchored Sheet-Pile Walls 460 9.9 Free Earth Support Method for Penetration of Sandy Soil 461 9.10 Design Charts for Free Earth Support Method (Penetration into Sandy Soil) 465 9.11 Moment Reduction for Anchored Sheet-Pile Walls 469 9.12 Computational Pressure Diagram Method for Penetration into Sandy Soil 472 9.13 Fixed Earth-Support Method for Penetration into Sandy Soil 476 9.14 Field Observations for Anchor Sheet Pile Walls 479 9.15 Free Earth Support Method for Penetration of Clay 482 9.16 Anchors 486 9.17 Holding Capacity of Anchor Plates in Sand 488 9.18 Holding Capacity of Anchor Plates in Clay (f 5 0 Condition) 495 9.19 Ultimate Resistance of Tiebacks 495 Problems 497 References 500 Contents 10 Braced Cuts 501 10.1 Introduction 501 10.2 Pressure Envelope for Braced-Cut Design 502 10.3 Pressure Envelope for Cuts in Layered Soil 506 10.4 Design of Various Components of a Braced Cut 507 10.5 Case Studies of Braced Cuts 515 10.6 Bottom Heave of a Cut in Clay 520 10.7 Stability of the Bottom of a Cut in Sand 524 10.8 Lateral Yielding of Sheet Piles and Ground Settlement 529 Problems 531 References 533 11 Pile Foundations 535 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 11.12 11.13 11.14 11.15 11.16 11.17 11.18 11.19 Introduction 535 Types of Piles and Their Structural Characteristics 537 Estimating Pile Length 546 Installation of Piles 548 Load Transfer Mechanism 551 Equations for Estimating Pile Capacity 554 Meyerhof’s Method for Estimating Qp 557 Vesic’s Method for Estimating Qp 560 Coyle and Castello’s Method for Estimating Qp in Sand 563 Correlations for Calculating Qp with SPT and CPT Results 567 Frictional Resistance (Qs) in Sand 568 Frictional (Skin) Resistance in Clay 575 Point Bearing Capacity of Piles Resting on Rock 579 Pile Load Tests 583 Elastic Settlement of Piles 588 Laterally Loaded Piles 591 Pile-Driving Formulas 606 Pile Capacity For Vibration-Driven Piles 611 Negative Skin Friction 613 Group Piles 617 11.20 Group Efficiency 617 11.21 Ultimate Capacity of Group Piles in Saturated Clay 621 11.22 Elastic Settlement of Group Piles 624 11.23 Consolidation Settlement of Group Piles 626 xiii xiv Contents 11.24 Piles in Rock 629 Problems 629 References 634 12 Drilled-Shaft Foundations 637 12.1 12.2 12.3 12.4 12.5 12.6 12.7 Introduction 637 Types of Drilled Shafts 638 Construction Procedures 639 Other Design Considerations 645 Load Transfer Mechanism 646 Estimation of Load-Bearing Capacity 646 Drilled Shafts in Granular Soil: Load-Bearing Capacity 648 12.8 Load-Bearing Capacity Based on Settlement 652 12.9 Drilled Shafts in Clay: Load-Bearing Capacity 661 12.10 Load-Bearing Capacity Based on Settlement 663 12.11 Settlement of Drilled Shafts at Working Load 668 12.12 Lateral Load-Carrying Capacity—Characteristic Load and Moment Method 670 12.13 Drilled Shafts Extending into Rock 679 Problems 681 References 685 13 Foundations on Difficult Soils 686 13.1 Introduction 686 Collapsible Soil 686 13.2 13.3 13.4 13.5 Definition and Types of Collapsible Soil 686 Physical Parameters for Identification 687 Procedure for Calculating Collapse Settlement 691 Foundation Design in Soils Not Susceptible to Wetting 692 13.6 Foundation Design in Soils Susceptible to Wetting 694 Expansive Soils 13.7 13.8 13.9 13.10 695 General Nature of Expansive Soils 695 Unrestrained Swell Test 699 Swelling Pressure Test 700 Classification of Expansive Soil on the Basis of Index Tests 705 Contents 13.11 Foundation Considerations for Expansive Soils 708 13.12 Construction on Expansive Soils 711 Sanitary Landfills 716 13.13 General Nature of Sanitary Landfills 716 13.14 Settlement of Sanitary Landfills 717 Problems 719 References 720 14 Soil Improvement and Ground Modification 722 14.1 Introduction 722 14.2 General Principles of Compaction 723 14.3 Field Compaction 727 14.4 Compaction Control for Clay Hydraulic Barriers 730 14.5 Vibroflotation 732 14.6 Blasting 739 14.7 Precompression 739 14.8 Sand Drains 745 14.9 Prefabricated Vertical Drains 756 14.10 Lime Stabilization 760 14.11 Cement Stabilization 764 14.12 Fly-Ash Stabilization 766 14.13 Stone Columns 767 14.14 Sand Compaction Piles 772 14.15 Dynamic Compaction 774 14.16 Jet Grouting 776 Problems 778 References 781 Answers to Selected Problems 783 Index 789 xv Preface Soil mechanics and foundation engineering have developed rapidly during the last fifty years. Intensive research and observation in the field and the laboratory have refined and improved the science of foundation design. Originally published in the fall of 1983 with a 1984 copyright, this text on the principles of foundation engineering is now in the seventh edition. The use of this text throughout the world has increased greatly over the years; it also has been translated into several languages. New and improved materials that have been published in various geotechnical engineering journals and conference proceedings have been incorporated into each edition of the text. Principles of Foundation Engineering is intended primarily for undergraduate civil engineering students. The first chapter, on Geotechnical Properties of Soil, reviews the topics covered in the introductory soil mechanics course, which is a prerequisite for the foundation engineering course. The text is composed of fourteen chapters with examples and problems, and an answer section for selected problems. The chapters are mostly devoted to the geotechnical aspects of foundation design. Both Systéime International (SI) units and English units are used in the text. Because the text introduces the application of fundamental concepts of foundation analysis and design to civil engineering students, the mathematical derivations are not always presented; instead, just the final form of the equation is given. A list of references for further information and study is included at the end of each chapter. Each chapter contains many example problems that will help students understand the application of the equations and graphs. For better understanding and visualization of the ideas and field practices, about thirty new photographs have been added in this edition. A number of practice problems also are given at the end of each chapter. Answers to some of these problems are given at the end of the text. The following is a brief overview of the changes from the sixth edition. • • • • In several parts of the text, the presentation has been thoroughly reorganized for better understanding. A number of new case studies have been added to familiarize students with the deviations from theory to practice. In Chapter 1 on Geotechnical Properties of Soil, new sections on liquidity index and activity have been added. The discussions on hydraulic conductivity of clay, relative density, and the friction angle of granular soils have been expanded. Expanded treatment of the weathering process of rocks is given in Chapter 2, Natural Soil Deposits and Subsoil Exploration. xvii xviii Preface • • • • • • • • • In Chapter 3 (Shallow Foundations: Ultimate Bearing Capacity), a new case study on bearing capacity failure in soft saturated clay has been added. Also included is the reduction factor method for estimating the ultimate bearing capacity of eccentrically loaded strip foundations on granular soil. Chapter 4, Ultimate Bearing Capacity of Shallow Foundations: Special Cases, has new sections on the ultimate bearing capacity of weaker soil underlain by a stronger soil, the seismic bearing capacity of foundations at the edge of a granular slope, foundations on rocks, and the stress characteristics solution for foundations located on the top of granular slopes. Stress distribution due to a point load and uniformly loaded circular and rectangular areas located on the surface of a Westergaard-type material has been added to Chapter 5 on Allowable Bearing Capacity and Settlement. Also included in this chapter is the procedure to estimate foundation settlement based on Pressuremeter test results. Lateral earth pressure due to a surcharge on unyielding retaining structures is now included in Chapter 7 (Lateral Earth Pressure). Also included in this chapter is the solution for passive earth pressure on a retaining wall with inclined back face and horizontal granular backfill using the method of triangular slices. Chapter 8 on Retaining Walls has a new case study. A more detailed discussion is provided on the design procedure for geogrid-reinforced retaining walls. Chapter 9 on Sheet Pile Walls has an added section on the holding capacity of plate anchors based on the stress characteristics solution. Two case studies have been added to the chapter on Braced Cuts (Chapter 10). The chapter on Pile Foundations (Chapter 11) has been thoroughly reorganized for better understanding. Based on recent publications, new recommendations have been made to estimate the load-bearing capacity of drilled shafts extending to rock (Chapter 12). As my colleagues in the geotechnical engineering area well know, foundation analysis and design is not just a matter of using theories, equations and graphs from a textbook. Soil profiles found in nature are seldom homogeneous, elastic, and isotropic. The educated judgment needed to properly apply the theories, equations, and graphs to the evaluation of soils, foundations, and foundation design cannot be overemphasized or completely taught in the classroom. Field experience must supplement classroom work. The following individuals were kind enough to share their photographs which have been included in this new edition. • • • • • • • • • • Professor A. S. Wayal, K. J. Somayia Polytechnic, Mumbai, India Professor Sanjeev Kumar, Southern Illinois University, Carbondale, Illinois Mr. Paul J. Koszarek, Professional Service Industries, Inc., Waukesha, Wisconsin Professor Khaled Sobhan, Florida Atlantic University, Boca Raton, Florida Professor Jean-Louis Briaud, Texas A&M University, College Station, Texas Dr. Dharma Shakya, Geotechnical Solutions, Inc., Irvine, California Mr. Jon Ridgeway, Tensar International, Atlanta, Georgia Professor N. Sivakugan, James Cook University, Townsville, Queensland, Australia Professor Anand J. Puppala, University of Texas at Arlington, Arlington, Texas Professor Thomas M. Petry, Missouri University of Science and Technology, Rolla, Missouri Preface xix Thanks are due to Neill Belk, graduate student at the University of North Carolina at Charlotte, and Jennifer Nicks, graduate student at Texas A&M University, College Station, Texas, for their help during the preparation of this revised edition. I am also grateful for several helpful suggestions of Professor Adel S. Saada of Case Western Reserve University, Cleveland, Ohio. Thanks are due to Chris Carson, Executive Director, Global Publishing Program; and Hilda Gowans, Senior Developmental Editor, Engineering, Cengage Learning; Lauren Betsos, Marketing Manager; and Rose Kernan of RPK Editorial Services for their interest and patience during the revision and production of the manuscript. For the past twenty-seven years, my primary source of inspiration has been the immeasurable energy of my wife, Janice. I am grateful for her continual help in the development of the original text and its six subsequent revisions. Braja M. Das 1 1.1 Geotechnical Properties of Soil Introduction The design of foundations of structures such as buildings, bridges, and dams generally requires a knowledge of such factors as (a) the load that will be transmitted by the superstructure to the foundation system, (b) the requirements of the local building code, (c) the behavior and stress-related deformability of soils that will support the foundation system, and (d) the geological conditions of the soil under consideration. To a foundation engineer, the last two factors are extremely important because they concern soil mechanics. The geotechnical properties of a soil—such as its grain-size distribution, plasticity, compressibility, and shear strength—can be assessed by proper laboratory testing. In addition, recently emphasis has been placed on the in situ determination of strength and deformation properties of soil, because this process avoids disturbing samples during field exploration. However, under certain circumstances, not all of the needed parameters can be or are determined, because of economic or other reasons. In such cases, the engineer must make certain assumptions regarding the properties of the soil. To assess the accuracy of soil parameters—whether they were determined in the laboratory and the field or whether they were assumed—the engineer must have a good grasp of the basic principles of soil mechanics. At the same time, he or she must realize that the natural soil deposits on which foundations are constructed are not homogeneous in most cases. Thus, the engineer must have a thorough understanding of the geology of the area—that is, the origin and nature of soil stratification and also the groundwater conditions. Foundation engineering is a clever combination of soil mechanics, engineering geology, and proper judgment derived from past experience. To a certain extent, it may be called an art. When determining which foundation is the most economical, the engineer must consider the superstructure load, the subsoil conditions, and the desired tolerable settlement. In general, foundations of buildings and bridges may be divided into two major categories: (1) shallow foundations and (2) deep foundations. Spread footings, wall footings, and mat foundations are all shallow foundations. In most shallow foundations, the depth of embedment can be equal to or less than three to four times the width of the foundation. Pile and drilled shaft foundations are deep foundations. They are used when top layers have poor 1 2 Chapter 1: Geotechnical Properties of Soil load-bearing capacity and when the use of shallow foundations will cause considerable structural damage or instability. The problems relating to shallow foundations and mat foundations are considered in Chapters 3, 4, 5, and 6. Chapter 11 discusses pile foundations, and Chapter 12 examines drilled shafts. This chapter serves primarily as a review of the basic geotechnical properties of soils. It includes topics such as grain-size distribution, plasticity, soil classification, effective stress, consolidation, and shear strength parameters. It is based on the assumption that you have already been exposed to these concepts in a basic soil mechanics course. 1.2 Grain-Size Distribution In any soil mass, the sizes of the grains vary greatly. To classify a soil properly, you must know its grain-size distribution. The grain-size distribution of coarse-grained soil is generally determined by means of sieve analysis. For a fine-grained soil, the grain-size distribution can be obtained by means of hydrometer analysis. The fundamental features of these analyses are presented in this section. For detailed descriptions, see any soil mechanics laboratory manual (e.g., Das, 2009). Sieve Analysis A sieve analysis is conducted by taking a measured amount of dry, well-pulverized soil and passing it through a stack of progressively finer sieves with a pan at the bottom. The amount of soil retained on each sieve is measured, and the cumulative percentage of soil passing through each is determined. This percentage is generally referred to as percent finer. Table 1.1 contains a list of U.S. sieve numbers and the corresponding size of their openings. These sieves are commonly used for the analysis of soil for classification purposes. Table 1.1 U.S. Standard Sieve Sizes Sieve No. Opening (mm) 4 6 8 10 16 20 30 40 50 60 80 100 140 170 200 270 4.750 3.350 2.360 2.000 1.180 0.850 0.600 0.425 0.300 0.250 0.180 0.150 0.106 0.088 0.075 0.053 1.2 Grain-Size Distribution 3 Percent finer (by weight) 100 80 60 40 20 0 10 1 0.1 Grain size, D (mm) 0.01 Figure 1.1 Grain-size distribution curve of a coarse-grained soil obtained from sieve analysis The percent finer for each sieve, determined by a sieve analysis, is plotted on semilogarithmic graph paper, as shown in Figure 1.1. Note that the grain diameter, D, is plotted on the logarithmic scale and the percent finer is plotted on the arithmetic scale. Two parameters can be determined from the grain-size distribution curves of coarsegrained soils: (1) the uniformity coefficient (Cu ) and (2) the coefficient of gradation, or coefficient of curvature (Cc ). These coefficients are D60 D10 (1.1) D230 (D60 ) (D10 ) (1.2) Cu 5 and Cc 5 where D10, D30, and D60 are the diameters corresponding to percents finer than 10, 30, and 60%, respectively. For the grain-size distribution curve shown in Figure 1.1, D10 5 0.08 mm, D30 5 0.17 mm, and D60 5 0.57 mm. Thus, the values of Cu and Cc are Cu 5 0.57 5 7.13 0.08 and Cc 5 0.172 5 0.63 (0.57) (0.08) 4 Chapter 1: Geotechnical Properties of Soil Parameters Cu and Cc are used in the Unified Soil Classification System, which is described later in the chapter. Hydrometer Analysis Hydrometer analysis is based on the principle of sedimentation of soil particles in water. This test involves the use of 50 grams of dry, pulverized soil. A deflocculating agent is always added to the soil. The most common deflocculating agent used for hydrometer analysis is 125 cc of 4% solution of sodium hexametaphosphate. The soil is allowed to soak for at least 16 hours in the deflocculating agent. After the soaking period, distilled water is added, and the soil–deflocculating agent mixture is thoroughly agitated. The sample is then transferred to a 1000-ml glass cylinder. More distilled water is added to the cylinder to fill it to the 1000-ml mark, and then the mixture is again thoroughly agitated. A hydrometer is placed in the cylinder to measure the specific gravity of the soil–water suspension in the vicinity of the instrument’s bulb (Figure 1.2), usually over a 24-hour period. Hydrometers are calibrated to show the amount of soil that is still in suspension at any given time t. The largest diameter of the soil particles still in suspension at time t can be determined by Stokes’ law, D5 18h L Å (Gs 2 1)gw Å t where D 5 diameter of the soil particle Gs 5 specific gravity of soil solids h 5 viscosity of water L Figure 1.2 Hydrometer analysis (1.3) 1.4 Weight–Volume Relationships 5 gw 5 unit weight of water L 5 effective length (i.e., length measured from the water surface in the cylinder to the center of gravity of the hydrometer; see Figure 1.2) t 5 time Soil particles having diameters larger than those calculated by Eq. (1.3) would have settled beyond the zone of measurement. In this manner, with hydrometer readings taken at various times, the soil percent finer than a given diameter D can be calculated and a grain-size distribution plot prepared. The sieve and hydrometer techniques may be combined for a soil having both coarse-grained and fine-grained soil constituents. 1.3 Size Limits for Soils Several organizations have attempted to develop the size limits for gravel, sand, silt, and clay on the basis of the grain sizes present in soils. Table 1.2 presents the size limits recommended by the American Association of State Highway and Transportation Officials (AASHTO) and the Unified Soil Classification systems (Corps of Engineers, Department of the Army, and Bureau of Reclamation). The table shows that soil particles smaller than 0.002 mm have been classified as clay. However, clays by nature are cohesive and can be rolled into a thread when moist. This property is caused by the presence of clay minerals such as kaolinite, illite, and montmorillonite. In contrast, some minerals, such as quartz and feldspar, may be present in a soil in particle sizes as small as clay minerals, but these particles will not have the cohesive property of clay minerals. Hence, they are called claysize particles, not clay particles. 1.4 Weight–Volume Relationships In nature, soils are three-phase systems consisting of solid soil particles, water, and air (or gas). To develop the weight–volume relationships for a soil, the three phases can be separated as shown in Figure 1.3a. Based on this separation, the volume relationships can then be defined. The void ratio, e, is the ratio of the volume of voids to the volume of soil solids in a given soil mass, or Vv e5 (1.4) Vs Table 1.2 Soil-Separate Size Limits Classification system Grain size (mm) Unified Gravel: 75 mm to 4.75 mm Sand: 4.75 mm to 0.075 mm Silt and clay (fines): ,0.075 mm AASHTO Gravel: 75 mm to 2 mm Sand: 2 mm to 0.05 mm Silt: 0.05 mm to 0.002 mm Clay: ,0.002 mm 6 Chapter 1: Geotechnical Properties of Soil Volume Note: Va + Vw + Vs = V Ww + Ws = W Volume Weight Air Va V Weight Vw Wa = 0 Ww W V Solid Vs Ws (a) Weight Volume Va Air V = e Note: Vw = wGs = Se Ww = wGsw Vw = wGs Vs = 1 Wa = 0 Solid Ws = Gsw (b) Unsaturated soil; Vs = 1 Weight Volume V = e Ww = wGsw = ew Vw = wGs = e Vs = 1 Solid Ws = Gsw (c) Saturated soil; Vs = 1 Figure 1.3 Weight–volume relationships where Vv 5 volume of voids Vs 5 volume of soil solids The porosity, n, is the ratio of the volume of voids to the volume of the soil specimen, or n5 where V 5 total volume of soil Vv V (1.5) 1.4 Weight–Volume Relationships 7 Moreover, n5 Vv Vs Vv Vv e 5 5 5 V V V Vs 1 Vv 11e s v 1 Vs Vs (1.6) The degree of saturation, S, is the ratio of the volume of water in the void spaces to the volume of voids, generally expressed as a percentage, or S(%) 5 Vw 3 100 Vv (1.7) where Vw 5 volume of water Note that, for saturated soils, the degree of saturation is 100%. The weight relationships are moisture content, moist unit weight, dry unit weight, and saturated unit weight, often defined as follows: Moisture content 5 w(%) 5 Ww 3 100 Ws (1.8) where Ws 5 weight of the soil solids Ww 5 weight of water Moist unit weight 5 g 5 W V (1.9) where W 5 total weight of the soil specimen 5 Ws 1 Ww The weight of air, Wa, in the soil mass is assumed to be negligible. Dry unit weight 5 gd 5 Ws V (1.10) When a soil mass is completely saturated (i.e., all the void volume is occupied by water), the moist unit weight of a soil [Eq. (1.9)] becomes equal to the saturated unit weight (gsat ). So g 5 gsat if Vv 5 Vw. More useful relations can now be developed by considering a representative soil specimen in which the volume of soil solids is equal to unity, as shown in Figure 1.3b. Note that if Vs 5 1, then, from Eq. (1.4), Vv 5 e, and the weight of the soil solids is Ws 5 Gsgw where Gs 5 specific gravity of soil solids gw 5 unit weight of water (9.81 kN>m3) 8 Chapter 1: Geotechnical Properties of Soil Also, from Eq. (1.8), the weight of water Ww 5 wWs. Thus, for the soil specimen under consideration, Ww 5 wWs 5 wGsgw. Now, for the general relation for moist unit weight given in Eq. (1.9), Ws 1 Ww Gsgw (1 1 w) W 5 5 V Vs 1 Vv 11e g5 (1.11) Similarly, the dry unit weight [Eq. (1.10)] is gd 5 Ws Ws Gsgw 5 5 V Vs 1 Vv 11e (1.12) From Eqs. (1.11) and (1.12), note that gd 5 g 11w (1.13) According to Eq. (1.7), degree of saturation is S5 Vw Vv Now, referring to Fig. 1.3(b), Vw 5 wGs and Vv 5 e Thus, S5 Vw wGs 5 e Vv (1.14) For a saturated soil, S ⫽ 1. So e 5 wGs (1.15) The saturated unit weight of soil then becomes gsat 5 Ws 1 Ww Gsgw 1 e gw 5 Vs 1 Vv 11e (1.16) 1.4 Weight–Volume Relationships 9 In SI units, Newton or kiloNewton is weight and is a derived unit, and g or kg is mass. The relationships given in Eqs. (1.11), (1.12) and (1.16) can be expressed as moist, dry, and saturated densities as follow: r5 Gsrw (1 1 w) 11e (1.17) Gsrw 11e (1.18) rd 5 r sat 5 rw (Gs 1 e) 11e (1.19) where , d, sat ⫽ moist density, dry density, and saturated density, respectively w ⫽ density of water (⫽ 1000 kg/m3) Relationships similar to Eqs. (1.11), (1.12), and (1.16) in terms of porosity can also be obtained by considering a representative soil specimen with a unit volume (Figure 1.3c). These relationships are g 5 Gsgw (1 2 n) (1 1 w) (1.20) gd 5 (1 2 n)Gsgw (1.21) gsat 5 3(1 2 n)Gs 1 n4gw (1.22) and Table 1.3 gives a summary of various forms of relationships that can be obtained for ˝, ˝d, and ˝sat. Table 1.3 Various Forms of Relationships for ␥, ␥d, and ␥sat Unit-weight relationship Dry unit weight (1 1 w)Gsgw 11e (Gs 1 Se)gw g5 11e (1 1 w)Gsgw g5 wGs 11 S ␥ ⫽ Gs␥w(1 ⫺ n)(1 + w) g 11w Gsgw gd 5 11e ␥d ⫽ Gs␥w(1 ⫺ n) Gs g gd 5 wGs w 11 S eSgw gd 5 (1 1 e)w g5 gd 5 ␥d ⫽ ␥sat ⫺ n␥w gd 5 gsat 2 a e bg 11e w Saturated unit weight (Gs 1 e)gw 11e ␥sat ⫽ [(1 ⫺ n)Gs + n]␥w gsat 5 gsat 5 a 11w bG g 1 1 wGs s w e 11w bg gsat 5 a b a w 11e w ␥sat ⫽ ␥d + n␥w gsat 5 gd 1 a e bg 11e w 10 Chapter 1: Geotechnical Properties of Soil Table 1.4 Specific Gravities of Some Soils Type of soil Gs Quartz sand Silt Clay Chalk Loess Peat 2.64–2.66 2.67–2.73 2.70–2.9 2.60–2.75 2.65–2.73 1.30–1.9 Except for peat and highly organic soils, the general range of the values of specific gravity of soil solids (Gs ) found in nature is rather small. Table 1.4 gives some representative values. For practical purposes, a reasonable value can be assumed in lieu of running a test. 1.5 Relative Density In granular soils, the degree of compaction in the field can be measured according to the relative density, defined as Dr (%) 5 emax 2 e 3 100 emax 2 emin (1.23) where emax 5 void ratio of the soil in the loosest state emin 5 void ratio in the densest state e 5 in situ void ratio The relative density can also be expressed in terms of dry unit weight, or Dr (%) 5 b gd 2 gd(min) gd(max) 2 gd(min) r gd(max) gd 3 100 (1.24) where gd 5 in situ dry unit weight gd(max) 5 dry unit weight in the densest state; that is, when the void ratio is emin gd(min) 5 dry unit weight in the loosest state; that is, when the void ratio is emax The denseness of a granular soil is sometimes related to the soil’s relative density. Table 1.5 gives a general correlation of the denseness and Dr. For naturally occurring sands, the magnitudes of emax and emin [Eq. (1.23)] may vary widely. The main reasons for such wide variations are the uniformity coefficient, Cu, and the roundness of the particles. 1.5 Relative Density 11 Table 1.5 Denseness of a Granular Soil Relative density, Dr (%) Description 0–20 20–40 40–60 60–80 80–100 Very loose Loose Medium Dense Very dense Example 1.1 The moist weight of 28.3 ⫻ 10⫺4 m3 of soil is 54.27 N. If the moisture content is 12% and the specific gravity of soil solids is 2.72, find the following: a. b. c. d. e. f. Moist unit weight (kN/m3) Dry unit weight (kN/m3) Void ratio Porosity Degree of saturation (%) Volume occupied by water (m3) Solution Part a From Eq. (1.9), g5 W 54.27 3 1023 5 5 19.18 kN>m3 V 0.00283 Part b From Eq. (1.13), gd 5 g 19.18 5 5 17.13 kN>m3 11w 12 11 100 Part c From Eq. (1.12), gd 5 or 17.13 5 Gsgw 11e (2.72) (9.81) 11e e 5 0.56 12 Chapter 1: Geotechnical Properties of Soil Part d From Eq. (1.6), n5 e 0.56 5 5 0.359 11e 1 1 0.56 Part e From Eq. (1.14), S5 wGs (0.12) (2.72) 5 0.583 5 e 0.56 Part f From Eq. (1.12), Ws 5 W 54.27 3 1023 5 5 48.46 3 1023 kN 11w 1.12 Ww 5 W 2 Ws 5 54.27 3 1023 2 48.46 3 1023 5 5.81 3 1023 kN Vw 5 5.81 3 1023 5 0.592 3 1023 m3 9.81 ■ Example 1.2 The dry density of a sand with a porosity of 0.387 is 1600 kg/m3. Find the void ratio of the soil and the specific gravity of the soil solids. Solution Void ratio Given: n ⫽ 0.387. From Eq. (1.6), n 0.387 e5 5 5 0.631 12n 1 2 0.387 Specific gravity of soil solids From Eq. (1.18), rd 5 1600 5 Gsrw 11e Gs (1000) 1.631 Gs 5 2.61 ■ 1.5 Relative Density 13 Example 1.3 The moist unit weight of a soil is 19.2 kN/m3. Given Gs ⫽ 2.69 and moisture content w 5 9.8%, determine a. b. c. d. Dry unit weight (kN/m3) Void ratio Porosity Degree of saturation (%) Solution Part a From Eq. (1.13), gd 5 g 5 11w 19.2 5 17.49 kN>m3 9.8 11 100 Part b From Eq. (1.12), gd 5 17.49 kN>m3 5 Gsgw (2.69) (9.81) 5 11e 11e e 5 0.509 Part c From Eq. (1.6), n5 e 0.509 5 5 0.337 11e 1 1 0.509 Part d From Eq. (1.14), S5 (0.098) (2.69) wGs 5 c d (100) 5 51.79% e 0.509 ■ Example 1.4 For a saturated soil, given w ⫽ 40% and Gs ⫽ 2.71, determine the saturated and dry unit weights in lb/ft3 and kN/m3. Solution For saturated soil, from Eq. (1.15), e ⫽ wGs ⫽ (0.4)(2.71) ⫽ 1.084 14 Chapter 1: Geotechnical Properties of Soil From Eq. (1.16), gsat 5 (Gs 1 e)gw (2.71 1 1.084)9.81 5 5 17.86 kN>m3 11e 1 1 1.084 From Eq. (1.12), gd 5 Gsgw (2.71) (9.81) 5 5 12.76 kN>m3 11e 1 1 1.084 ■ Example 1.5 The mass of a moist soil sample collected from the field is 465 grams, and its oven dry mass is 405.76 grams. The specific gravity of the soil solids was determined in the laboratory to be 2.68. If the void ratio of the soil in the natural state is 0.83, find the following: a. The moist density of the soil in the field (kg/m3) b. The dry density of the soil in the field (kg/m3) c. The mass of water, in kilograms, to be added per cubic meter of soil in the field for saturation Solution Part a From Eq. (1.8), w5 Ww 465 2 405.76 59.24 5 5 5 14.6% Ws 405.76 405.76 From Eq. (1.17), r5 Gsrw 1 wGsrw Gsrw (1 1 w) (2.68) (1000) (1.146) 5 5 11e 11e 1.83 5 1678.3 kg>m3 Part b From Eq. (1.18), rd 5 Gsrw (2.68) (1000) 5 5 1468.48 kg>m3 11e 1.83 1.6 Atterberg Limits 15 Part c Mass of water to be added ⫽ sat ⫺ From Eq. (1.19), rsat 5 Gsrw 1 erw rw (Gs 1 e) (1000) (2.68 1 0.83) 5 5 5 1918 kg>m3 11e 11e 1.83 So, mass of water to be added ⫽ 1918 ⫺ 1678.3 ⫽ 239.7 kg/m3. ■ Example 1.6 The maximum and minimum dry unit weights of a sand are 17.1 kN>m3 and 14.2 kN>m3, respectively. The sand in the field has a relative density of 70% with a moisture content of 8%. Determine the moist unit weight of the sand in the field. Solution From Eq. (1.24), gd 2 gd(min) dc gd(max) Dr 5 c gd(max) 2 gd(min) 0.7 5 c gd 2 14.2 17.1 dc d gd 17.1 2 14.2 gd d gd 5 16.11 kN>m3 g 5 gd (1 1 w) 5 16.11a1 1 1.6 8 b 5 17.4 kN , m3 100 ■ Atterberg Limits When a clayey soil is mixed with an excessive amount of water, it may flow like a semiliquid. If the soil is gradually dried, it will behave like a plastic, semisolid, or solid material, depending on its moisture content. The moisture content, in percent, at which the soil changes from a liquid to a plastic state is defined as the liquid limit (LL). Similarly, the moisture content, in percent, at which the soil changes from a plastic to a semisolid state and from a semisolid to a solid state are defined as the plastic limit (PL) and the shrinkage limit (SL), respectively. These limits are referred to as Atterberg limits (Figure 1.4): • • The liquid limit of a soil is determined by Casagrande’s liquid device (ASTM Test Designation D-4318) and is defined as the moisture content at which a groove closure of 12.7 mm occurs at 25 blows. The plastic limit is defined as the moisture content at which the soil crumbles when rolled into a thread of 3.18 mm in diameter (ASTM Test Designation D-4318). 16 Chapter 1: Geotechnical Properties of Soil Solid state Semisolid state Plastic state Semiliquid state Increase of moisture content Volume of the soil–water mixture SL PL LL Moisture content Figure 1.4 Definition of Atterberg limits • The shrinkage limit is defined as the moisture content at which the soil does not undergo any further change in volume with loss of moisture (ASTM Test Designation D-427). The difference between the liquid limit and the plastic limit of a soil is defined as the plasticity index (PI), or PI 5 LL 2 PL 1.7 (1.25) Liquidity Index The relative consistency of a cohesive soil in the natural state can be defined by a ratio called the liquidity index, which is given by LI 5 w 2 PL LL 2 PL (1.26) where w ⫽ in situ moisture content of soil. The in situ moisture content for a sensitive clay may be greater than the liquid limit. In this case, LI ⬎ 1 These soils, when remolded, can be transformed into a viscous form to flow like a liquid. 1.9 Soil Classification Systems 17 Soil deposits that are heavily overconsolidated may have a natural moisture content less than the plastic limit. In this case, LI ⬍ 0 1.8 Activity Because the plasticity of soil is caused by the adsorbed water that surrounds the clay particles, we can expect that the type of clay minerals and their proportional amounts in a soil will affect the liquid and plastic limits. Skempton (1953) observed that the plasticity index of a soil increases linearly with the percentage of clay-size fraction (% finer than 2 m by weight) present. The correlations of PI with the clay-size fractions for different clays plot separate lines. This difference is due to the diverse plasticity characteristics of the various types of clay minerals. On the basis of these results, Skempton defined a quantity called activity, which is the slope of the line correlating PI and % finer than 2 m. This activity may be expressed as A5 PI (% of clay-size fraction, by weight) (1.27) Activity is used as an index for identifying the swelling potential of clay soils. Typical values of activities for various clay minerals are given in Table 1.6. 1.9 Soil Classification Systems Soil classification systems divide soils into groups and subgroups based on common engineering properties such as the grain-size distribution, liquid limit, and plastic limit. The two major classification systems presently in use are (1) the American Association of State Highway and Transportation Officials (AASHTO) System and (2) the Unified Soil Classification System (also ASTM). The AASHTO system is used mainly for the classification of highway subgrades. It is not used in foundation construction. Table 1.6 Activities of Clay Minerals Mineral Smectites Illite Kaolinite Halloysite (4H2O) Halloysite (2H2O) Attapulgite Allophane Activity (A) 1–7 0.5–1 0.5 0.5 0.1 0.5–1.2 0.5–1.2 18 Chapter 1: Geotechnical Properties of Soil AASHTO System The AASHTO Soil Classification System was originally proposed by the Highway Research Board’s Committee on Classification of Materials for Subgrades and Granular Type Roads (1945). According to the present form of this system, soils can be classified according to eight major groups, A-1 through A-8, based on their grain-size distribution, liquid limit, and plasticity indices. Soils listed in groups A-1, A-2, and A-3 are coarse-grained materials, and those in groups A-4, A-5, A-6, and A-7 are fine-grained materials. Peat, muck, and other highly organic soils are classified under A-8. They are identified by visual inspection. The AASHTO classification system (for soils A-1 through A-7) is presented in Table 1.7. Note that group A-7 includes two types of soil. For the A-7-5 type, the plasticity Table 1.7 AASHTO Soil Classification System Granular materials (35% or less of total sample passing No. 200 sieve) General classification A-1 Group classification Sieve analysis (% passing) No. 10 sieve No. 40 sieve No. 200 sieve For fraction passing No. 40 sieve Liquid limit (LL) Plasticity index (PI) Usual type of material A-2 A-1-a A-1-b A-3 A-2-4 A-2-5 A-2-6 A-2-7 50 max 30 max 15 max 50 max 25 max 51 min 10 max 35 max 35 max 35 max 35 max Nonplastic Fine sand 40 max 41 min 40 max 41 min 10 max 10 max 11 min 11 min Silty or clayey gravel and sand 6 max Stone fragments, gravel, and sand Subgrade rating Excellent to good Silt–clay materials (More than 35% of total sample passing No. 200 sieve) General classification Group classification A-4 A-5 A-6 A-7 A-7-5a A-7-6b Sieve analysis (% passing) No. 10 sieve No. 40 sieve No. 200 sieve For fraction passing No. 40 sieve Liquid limit (LL) Plasticity index (PI) Usual types of material Subgrade rating a If PI < LL 2 30, the classification is A-7-5. If PI . LL 2 30, the classification is A-7-6. b 36 min 36 min 36 min 40 max 41 min 10 max 10 max Mostly silty soils 36 min 40 max 41 min 11 min 11 min Mostly clayey soils Fair to poor 1.9 Soil Classification Systems 19 index of the soil is less than or equal to the liquid limit minus 30. For the A-7-6 type, the plasticity index is greater than the liquid limit minus 30. For qualitative evaluation of the desirability of a soil as a highway subgrade material, a number referred to as the group index has also been developed. The higher the value of the group index for a given soil, the weaker will be the soil’s performance as a subgrade. A group index of 20 or more indicates a very poor subgrade material. The formula for the group index is GI 5 (F200 2 35) 30.2 1 0.005(LL 2 40) 4 1 0.01(F200 2 15) (PI 2 10) (1.28) where F200 5 percent passing No. 200 sieve, expressed as a whole number LL 5 liquid limit PI 5 plasticity index When calculating the group index for a soil belonging to group A-2-6 or A-2-7, use only the partial group-index equation relating to the plasticity index: GI 5 0.01(F200 2 15) (PI 2 10) (1.29) The group index is rounded to the nearest whole number and written next to the soil group in parentheses; for example, we have A-4 ()* Z Soil group (5) ()* Group index The group index for soils which fall in groups A-1-a, A-1-b, A-3, A-2-4, and A-2-5 is always zero. Unified System The Unified Soil Classification System was originally proposed by A. Casagrande in 1942 and was later revised and adopted by the United States Bureau of Reclamation and the U.S. Army Corps of Engineers. The system is currently used in practically all geotechnical work. In the Unified System, the following symbols are used for identification: Symbol G S M Description Gravel Sand Silt C O Pt H L W Clay Organic silts and clay Peat and highly organic soils High Low Well plasticity plasticity graded P Poorly graded 20 Chapter 1: Geotechnical Properties of Soil 70 Plasticity index, PI 60 U-line PI 0.9 (LL 8) 50 40 CL or OL 30 20 CL ML ML or OL 10 0 0 10 20 30 CH or OH A-line PI 0.73 (LL 20) MH or OH 40 50 60 70 Liquid limit, LL 80 90 100 Figure 1.5 Plasticity chart The plasticity chart (Figure 1.5) and Table 1.8 show the procedure for determining the group symbols for various types of soil. When classifying a soil be sure to provide the group name that generally describes the soil, along with the group symbol. Figures 1.6, 1.7, and 1.8 give flowcharts for obtaining the group names for coarse-grained soil, inorganic finegrained soil, and organic fine-grained soil, respectively. Example 1.7 Classify the following soil by the AASHTO classification system. Percent passing No. 4 sieve 5 92 Percent passing No. 10 sieve 5 87 Percent passing No. 40 sieve 5 65 Percent passing No. 200 sieve 5 30 Liquid limit 5 22 Plasticity index 5 8 Solution Table 1.7 shows that it is a granular material because less than 35% is passing a No. 200 sieve. With LL 5 22 (that is, less than 40) and PI 5 8 (that is, less than 10), the soil falls in group A-2-4. The soil is A-2-4(0). ■ 21 b Cu 5 D60>D10 Cc 5 (D30 ) 2 j , 0.75 PT OH Organic silt k, l, m, q Peat Organic clay k, l, m, p Elastic silt k , l , m Fat clay k , l , m Organic silt k, l, m, o Organic clay k, l, m, n Siltk , l , m Lean clayk , l , m Clayey sandg , h , i Silty sandg , h , i Poorly graded sandi Well-graded sandi Clayey gravel f , g , h Silty gravel f , g , h Poorly graded gravel f Well-graded gravel f k If soil contains 15 to 29% plus No. 200, add “with sand” or “with gravel,” whichever is predominant. l If soil contains >30% plus No. 200, predominantly sand, add “sandy” to group name. m If soil contains >30% plus No. 200, predominantly gravel, add “gravelly” to group name. n PI > 4 and plots on or above “A” line. o PI , 4 or plots below “A” line. p PI plots on or above “A” line. q PI plots below “A” line. Liquid limit—not dried Liquid limit—oven dried MH PI plots below “A” line OL CH , 0.75 ML CL PI plots on or above “A” line Liquid limit—not dried Liquid limit—oven dried PI , 4 or plots below “A” line j PI . 7 and plots on or above “A” line SC SP Cu , 6 and/or 1 . Cc . 3e Fines classify as CL or CH SW SM GC Fines classify as CL or CH Cu > 6 and 1 < Cc < 3e Fines classify as ML or MH GM GP Cu , 4 and/or 1 . Cc . 3e Fines classify as ML or MH GW Cu > 4 and 1 < Cc < 3e Primarily organic matter, dark in color, and organic odor Organic Inorganic Organic Inorganic Sand with Fines More than 12% finesd Clean Sands Less than 5% finesd Gravels with Fines More than 12% finesc Clean Gravels Less than 5% finesc D10 3 D60 f If soil contains >15% sand, add “with sand” to group name. g If fines classify as CL-ML, use dual symbol GC-GM or SC-SM. h If fines are organic, add “with organic fines” to group name. i If soil contains >15% gravel, add “with gravel” to group name. j If Atterberg limits plot in hatched area, soil is a CL-ML, silty clay. e Silts and Clays Liquid limit 50 or more Silts and Clays Liquid limit less than 50 Sands 50% or more of coarse fraction passes No. 4 sieve Gravels More than 50% of coarse fraction retained on No. 4 sieve Based on the material passing the 75-mm. (3-in) sieve. If field sample contained cobbles or boulders, or both, add “with cobbles or boulders, or both” to group name. c Gravels with 5 to 12% fines require dual symbols: GWGM well-graded gravel with silt; GW-GC well-graded gravel with clay; GP-GM poorly graded gravel with silt; GP-GC poorly graded gravel with clay. d Sands with 5 to 12% fines require dual symbols: SWSM well-graded sand with silt; SW-SC well-graded sand with clay; SP-SM poorly graded sand with silt; SP-SC poorly graded sand with clay. a Highly organic soils Fine-grained soils 50% or more passes the No. 200 sieve Coarse-grained soils More than 50% retained on No. 200 sieve Criteria for assigning group symbols and group names using laboratory testsa Soil classification Group symbol Group nameb Table 1.8 Unified Soil Classification Chart (after ASTM, 2009) (ASTM D2487-98: Standard Practice for Classification of Soils for Engineering Purposes (Unified Soil Classification). Copyright ASTM INTERNATIONAL. Reprinted with permission.) 22 12% fines 5–12% fines 5% fines 12% fines 5–12% fines fines ML or MH GC GC-GM fines CL-ML Cu 6 and/or 1 Cc 3 SW-SM SM SC SC-SM fines CL or CH fines CL-ML SP-SC SP-SM SW-SC fines ML or MH fines CL, CH (or CL-ML) fines ML or MH fines CL, CH (or CL-ML) SP Cu 6 and/or 1 Cc 3 Cu 6 and 1 Cc 3 SW fines ML or MH GM fines CL or CH GP-GC GP-GM fines ML or MH fines CL, CH (or CL-ML) fines ML or MH GW-GM GW-GC Cu 6 and 1 Cc 3 Cu 4 and/or 1 Cc 3 fines CL, CH (or CL-ML) GP Cu 4 and/or 1 Cc 3 Cu 4 and 1 Cc 3 GW Cu 4 and 1 Cc 3 Silty gravel Silty gravel with sand Clayey gravel Clayey gravel with sand Silty, clayey gravel Silty, clayey gravel with sand 15% sand 15% sand 15% sand 15% sand 15% sand 15% sand Well-graded sand with silt Well-graded sand with silt and gravel Well-graded sand with clay (or silty clay) Well-graded sand with clay and gravel (or silty clay and gravel) Poorly graded sand with silt Poorly graded sand with silt and gravel Poorly graded sand with clay (or silty clay) Poorly graded sand with clay and gravel (or silty clay and gravel) Silty sand Silty sand with gravel Clayey sand Clayey sand with gravel Silty, clayey sand Silty, clayey sand with gravel 15% gravel 15% gravel 15% gravel 15% gravel 15% gravel 15% gravel 15% gravel 15% gravel 15% gravel 15% gravel 15% gravel 15% gravel 15% gravel 15% gravel Well-graded sand Well-graded sand with gravel Poorly graded sand Poorly graded sand with gravel Poorly graded gravel with silt Poorly graded gravel with silt and sand Poorly graded gravel with clay (or silty clay) Poorly graded gravel with clay and sand (or silty clay and sand) 15% sand 15% sand 15% sand 15% sand 15% gravel 15% gravel 15% gravel 15% gravel Well-graded gravel with silt Well-graded gravel with silt and sand Well-graded gravel with clay (or silty clay) Well-graded gravel with clay and sand (or silty clay and sand) Well-graded gravel Well-graded gravel with sand Poorly graded gravel Poorly graded gravel with sand 15% sand 15% sand 15% sand 15% sand 15% sand 15% sand 15% sand 15% sand Group Name Figure 1.6 Flowchart for classifying coarse-grained soils (more than 50% retained on No. 200 Sieve) (After ASTM, 2009) (ASTM D2487-98: Standard Practice for Classification of Soils for Engineering Purposes (Unified Soil Classification). Copyright ASTM INTERNATIONAL. Reprinted with permission.) Sand % sand % gravel Gravel % gravel % sand 5% fines Group Symbol 23 Organic Inorganic Organic ( ( LL—ovendried 0.75 LL—not dried PI plots below “A”—line PI plots on or above “A”—line LL—ovendried 0.75 LL—not dried PI 4 or plots below “A”—line ) ) OH MH CH OL ML CL-ML CL See Figure 1.8 30% plus No. 200 30% plus No. 200 30% plus No. 200 30% plus No. 200 See figure 1.8 30% plus No. 200 30% plus No. 200 30% plus No. 200 30% plus No. 200 30% plus No. 200 30% plus No. 200 % sand % gravel % sand % gravel 15% plus No. 200 15-29% plus No. 200 % sand % gravel % sand % gravel 15% plus No. 200 15 −29% plus No. 200 % sand % gravel % sand % gravel 15% plus No. 200 15−29% plus No. 200 % sand % gravel % sand % gravel 15% plus No. 200 15−29% plus No. 200 % sand % gravel % sand % gravel 15% plus No. 200 15−29% plus No. 200 % sand % gravel % sand % gravel 15% gravel 15% gravel 15% sand 15% sand % sand % gravel % sand % gravel 15% gravel 15% gravel 15% sand 15% sand % sand % gravel % sand % gravel 15% gravel 15% gravel 15% sand 15% sand % sand % gravel % sand % gravel 15% gravel 15% gravel 15% sand 15% sand % sand % gravel % sand % gravel 15% gravel 15% gravel 15% sand 15% sand Elastic silt Elastic silt with sand Elastic silt with gravel Sandy elastic silt Sandy elastic silt with gravel Gravelly elastic silt Gravelly elastic silt with sand Fat clay Fat clay with sand Fat clay with gravel Sandy fat clay Sandy fat clay with gravel Gravelly fat clay Gravelly fat clay with sand Silt Silt with clay Silt with gravel Sandy silt Sandy silt with gravel Gravelly silt Gravelly silt with sand Silty clay Silty clay with sand Silty clay with gravel Sandy silty clay Sandy silty clay with gravel Gravelly silty clay Gravelly silty clay with sand Lean clay Lean clay with sand Lean clay with gravel Sandy lean clay Sandy lean clay with gravel Gravelly lean clay Gravelly lean clay with sand Group Name Figure 1.7 Flowchart for classifying fine-grained soil (50% or more passes No. 200 Sieve) (After ASTM, 2009)(ASTM D2487-98: Standard Practice for Classification of Soils for Engineering Purposes (Unified Soil Classification). Copyright ASTM INTERNATIONAL. Reprinted with permission.) LL 50 LL 50 Inorganic 4 PI 7 and plots on or above “A”—line PI 7 and plots on or above A ì ”—line Group Symbol 24 Plots below “A”—line Plots on or above “A”—line PI 4 and plots below “A”—line PI 4 and plots on or above “A”—line 30% plus No. 200 30% plus No. 200 30% plus No. 200 30% plus No. 200 30% plus No. 200 30% plus No. 200 30% plus No. 200 30% plus No. 200 % sand % gravel % sand % gravel 15% plus No. 200 15−29% plus No. 200 % sand % gravel % sand % gravel 15% plus No. 200 15−29% plus No. 200 % sand % gravel % sand % gravel 15% plus No. 200 15−29% plus No. 200 % sand % gravel % sand % gravel 15% plus No. 200 15−29% plus No. 200 % sand % gravel % sand % gravel 15% gravel 15% gravel 15% sand 15% sand % sand % gravel % sand % gravel 15% gravel 15% gravel 15% sand 15% sand % sand % gravel % sand % gravel 15% gravel 15% gravel 15% sand 15% sand % sand % gravel % sand % gravel 15% gravel 15% gravel 15% sand 15% sand Organic silt Organic silt with sand Organic silt with gravel Sandy organic silt Sandy organic silt with gravel Gravelly organic silt Gravelly organic silt with sand Organic clay Organic clay with sand Organic clay with gravel Sandy organic clay Sandy organic clay with gravel Gravelly organic clay Gravelly organic clay with sand Organic silt Organic silt with sand Organic silt with gravel Sandy organic silt Sandy organic silt with gravel Gravelly organic silt Gravelly organic silt with sand Organic clay Organic clay with sand Organic clay with gravel Sandy organic clay Sandy organic clay with gravel Gravelly organic clay Gravelly organic clay with sand Group Name Figure 1.8 Flowchart for classifying organic fine-grained soil (50% or more passes No. 200 Sieve) (After ASTM, 2009) (ASTM D2487-98: Standard Practice for Classification of Soils for Engineering Purposes (Unified Soil Classification). Copyright ASTM INTERNATIONAL. Reprinted with permission.) OH OL Group Symbol F 1.10 Hydraulic Conductivity of Soil 25 Example 1.8 Classify the following soil by the Unified Soil Classification System: Percent passing No. 4 sieve 5 82 Percent passing No. 10 sieve 5 71 Percent passing No. 40 sieve 5 64 Percent passing No. 200 sieve 5 41 Liquid limit 5 31 Plasticity index 5 12 Solution We are given that F200 5 41, LL 5 31, and PI 5 12. Since 59% of the sample is retained on a No. 200 sieve, the soil is a coarse-grained material. The percentage passing a No. 4 sieve is 82, so 18% is retained on No. 4 sieve (gravel fraction). The coarse fraction passing a No. 4 sieve (sand fraction) is 59 2 18 5 41% (which is more than 50% of the total coarse fraction). Hence, the specimen is a sandy soil. Now, using Table 1.8 and Figure 1.5, we identify the group symbol of the soil as SC. Again from Figure 1.6, since the gravel fraction is greater than 15%, the group name is clayey sand with gravel. ■ 1.10 Hydraulic Conductivity of Soil The void spaces, or pores, between soil grains allow water to flow through them. In soil mechanics and foundation engineering, you must know how much water is flowing through a soil per unit time. This knowledge is required to design earth dams, determine the quantity of seepage under hydraulic structures, and dewater foundations before and during their construction. Darcy (1856) proposed the following equation (Figure 1.9) for calculating the velocity of flow of water through a soil: v 5 ki (1.30) In this equation, v 5 Darcy velocity (unit: cm>sec) k 5 hydraulic conductivity of soil (unit: cm>sec) i 5 hydraulic gradient The hydraulic gradient is defined as i5 Dh L where Dh 5 piezometric head difference between the sections at AA and BB L 5 distance between the sections at AA and BB (Note: Sections AA and BB are perpendicular to the direction of flow.) (1.31) 26 Chapter 1: Geotechnical Properties of Soil h A B Direction of flow Soil A Direction of flow L B Figure 1.9 Definition of Darcy’s law Darcy’s law [Eq. (1.30)] is valid for a wide range of soils. However, with materials like clean gravel and open-graded rockfills, the law breaks down because of the turbulent nature of flow through them. The value of the hydraulic conductivity of soils varies greatly. In the laboratory, it can be determined by means of constant-head or falling-head permeability tests. The constanthead test is more suitable for granular soils. Table 1.9 provides the general range for the values of k for various soils. In granular soils, the value depends primarily on the void ratio. In the past, several equations have been proposed to relate the value of k to the void ratio in granular soil. However the author recommends the following equation for use (also see Carrier, 2003): e3 k~ (1.32) 11e where k 5 hydraulic conductivity e 5 void ratio Chapuis (2004) proposed an empirical relationship for k in conjunction with Eq. (1.32) as k(cm>s) 5 2.4622 cD210 0.7825 e3 d (1 1 e) where D 5 effective size (mm). Table 1.9 Range of the Hydraulic Conductivity for Various Soils Type of soil Medium to coarse gravel Coarse to fine sand Fine sand, silty sand Silt, clayey silt, silty clay Clays Hydraulic conductivity, k (cm/sec) Greater than 10 21 10 21 to 10 23 10 23 to 10 25 10 24 to 10 26 10 27 or less (1.33) 1.10 Hydraulic Conductivity of Soil 27 The preceding equation is valid for natural, uniform sand and gravel to predict k that is in the range of 1021 to 1023 cm>s. This can be extended to natural, silty sands without plasticity. It is not valid for crushed materials or silty soils with some plasticity. Based on laboratory experimental results, Amer and Awad (1974) proposed the following relationship for k in granular soil: k 5 3.5 3 1024 a rw e3 2.32 b bC 0.6 u D 10 a h 11e (1.34) where k is in cm/sec Cu ⫽ uniformity coefficient D10 ⫽ effective size (mm) w ⫽ density of water (g/cm3) ⫽ viscosity (g⭈s/cm2) At 20°C, w ⫽ 1 g/cm3 and ⬇ 0.1 ⫻ 10⫺4 g⭈s/cm2. So k 5 3.5 3 1024 a e3 1 bC 0.6D2.32 a b 1 1 e u 10 0.1 3 1024 or k (cm>sec) 5 35a e3 bC 0.6D2.32 1 1 e u 10 (1.35) Hydraulic Conductivity of Cohesive Soil According to their experimental observations, Samarasinghe, Huang, and Drnevich (1982) suggested that the hydraulic conductivity of normally consolidated clays could be given by the equation k5C en 11e (1.36) where C and n are constants to be determined experimentally. Some other empirical relationships for estimating the hydraulic conductivity in clayey soils are given in Table 1.10. One should keep in mind, however, that any empirical Table 1.10 Empirical Relationships for Estimating Hydraulic Conductivity in Clayey Soil Type of soil Source Relationshipa Clay Mesri and Olson (1971) log k ⫽ A⬘ log e + B⬘ e0 2 e log k 5 log k0 2 Ck Ck ⬇ 0.5e0 Taylor (1948) k0 ⫽ in situ hydraulic conductivity at void ratio e0 k ⫽ hydraulic conductivity at void ratio e Ck ⫽ hydraulic conductivity change index a 28 Chapter 1: Geotechnical Properties of Soil PI + CF = 1.25 2.8 1.0 2.4 Void ratiuo, e 2.0 0.75 1.6 1.2 0.5 0.8 0.4 10–11 10–10 10–9 5 × 10–9 k (m/sec) Figure 1.10 Variation of void ratio with hydraulic conductivity of clayey soils (Based on Tavenas, et al., 1983) relationship of this type is for estimation only, because the magnitude of k is a highly variable parameter and depends on several factors. Tavenas, et al. (1983) also gave a correlation between the void ratio and the hydraulic conductivity of clayey soil. This correlation is shown in Figure 1.10. An important point to note, however, is that in Figure 6.10, PI, the plasticity index, and CF, the claysize fraction in the soil, are in fraction (decimal) form. 1.11 Steady-State Seepage For most cases of seepage under hydraulic structures, the flow path changes direction and is not uniform over the entire area. In such cases, one of the ways of determining the rate of seepage is by a graphical construction referred to as the flow net, a concept based on Laplace’s theory of continuity. According to this theory, for a steady flow condition, the flow at any point A (Figure 1.11) can be represented by the equation kx '2h '2h '2h 1 k 1 k 50 y z 'x2 'y2 'z2 (1.37) where kx, ky, kz 5 hydraulic conductivity of the soil in the x, y, and z directions, respectively h 5 hydraulic head at point A (i.e., the head of water that a piezometer placed at A would show with the downstream water level as datum, as shown in Figure 1.11) For a two-dimensional flow condition, as shown in Figure 1.11, '2h 50 '2y 1.11 Steady-State Seepage Water level 29 Piezometers hmax h O O Flow line B D C A Water level D x Permeable soil layer y Equipotential line F E Rock z Figure 1.11 Steady-state seepage so Eq. (1.37) takes the form kx '2h '2h 1 k 50 z 'x2 'z2 (1.38) If the soil is isotropic with respect to hydraulic conductivity, kx 5 kz 5 k, and '2h '2h 1 50 'x2 'z2 (1.39) Equation (1.39), which is referred to as Laplace’s equation and is valid for confined flow, represents two orthogonal sets of curves known as flow lines and equipotential lines. A flow net is a combination of numerous equipotential lines and flow lines. A flow line is a path that a water particle would follow in traveling from the upstream side to the downstream side. An equipotential line is a line along which water, in piezometers, would rise to the same elevation. (See Figure 1.11.) In drawing a flow net, you need to establish the boundary conditions. For example, in Figure 1.11, the ground surfaces on the upstream (OOr) and downstream (DDr) sides are equipotential lines. The base of the dam below the ground surface, OrBCD, is a flow line. The top of the rock surface, EF, is also a flow line. Once the boundary conditions are established, a number of flow lines and equipotential lines are drawn by trial and error so that all the flow elements in the net have the same length-to-width ratio (L>B). In most cases, L>B is held to unity, that is, the flow elements are drawn as curvilinear “squares.” This method is illustrated by the flow net shown in Figure 1.12. Note that all flow lines must intersect all equipotential lines at right angles. Once the flow net is drawn, the seepage (in unit time per unit length of the structure) can be calculated as q 5 khmax Nf Nd n (1.40) 30 Chapter 1: Geotechnical Properties of Soil Water level hmax Water level B Permeable soil layer kx kz L Rock Figure 1.12 Flow net where Nf 5 number of flow channels Nd 5 number of drops n 5 width-to-length ratio of the flow elements in the flow net (B>L) hmax 5 difference in water level between the upstream and downstream sides The space between two consecutive flow lines is defined as a flow channel, and the space between two consecutive equipotential lines is called a drop. In Figure 1.12, Nf 5 2, Nd 5 7, and n 5 1. When square elements are drawn in a flow net, q 5 khmax 1.12 Nf Nd (1.41) Effective Stress The total stress at a given point in a soil mass can be expressed as s 5 sr 1 u (1.42) where s 5 total stress sr 5 effective stress u 5 pore water pressure The effective stress, sr, is the vertical component of forces at solid-to-solid contact points over a unit cross-sectional area. Referring to Figure 1.13a, at point A s 5 gh1 1 gsath2 u 5 h2gw where gw 5 unit weight of water gsat 5 saturated unit weight of soil 1.12 Effective Stress 31 h Water level Unit weight = h1 h2 B Groundwater level h1 Water Saturated unit weight = sat A h2 Saturated unit weight = sat A F2 F1 X Flow of water (a) (b) Figure 1.13 Calculation of effective stress So sr 5 (gh1 1 gsath2 ) 2 (h2gw ) 5 gh1 1 h2 (gsat 2 gw ) 5 gh1 1 grh2 (1.43) where gr 5 effective or submerged unit weight of soil. For the problem in Figure 1.13a, there was no seepage of water in the soil. Figure 1.13b shows a simple condition in a soil profile in which there is upward seepage. For this case, at point A, s 5 h1gw 1 h2gsat and u 5 (h1 1 h2 1 h)gw Thus, from Eq. (1.42), sr 5 s 2 u 5 (h1gw 1 h2gsat ) 2 (h1 1 h2 1 h)gw 5 h2 (gsat 2 gw ) 2 hgw 5 h2gr 2 hgw or sr 5 h2 ¢gr 2 h g ≤ 5 h2 (gr 2 igw ) h2 w (1.44) Note in Eq. (1.44) that h>h2 is the hydraulic gradient i. If the hydraulic gradient is very high, so that gr 2 igw becomes zero, the effective stress will become zero. In other words, there is no contact stress between the soil particles, and the soil will break up. This situation is referred to as the quick condition, or failure by heave. So, for heave, i 5 icr 5 gr Gs 2 1 5 gw 11e (1.45) 32 Chapter 1: Geotechnical Properties of Soil where icr 5 critical hydraulic gradient. For most sandy soils, icr ranges from 0.9 to 1.1, with an average of about unity. 1.13 Consolidation In the field, when the stress on a saturated clay layer is increased—for example, by the construction of a foundation—the pore water pressure in the clay will increase. Because the hydraulic conductivity of clays is very small, some time will be required for the excess pore water pressure to dissipate and the increase in stress to be transferred to the soil skeleton. According to Figure 1.14, if Ds is a surcharge at the ground surface over a very large area, the increase in total stress at any depth of the clay layer will be equal to Ds. However, at time t 5 0 (i.e., immediately after the stress is applied), the excess pore water pressure at any depth Du will equal Ds, or Du 5 Dhigw 5 Ds (at time t 5 0) Hence, the increase in effective stress at time t 5 0 will be Dsr 5 Ds 2 Du 5 0 Theoretically, at time t 5 `, when all the excess pore water pressure in the clay layer has dissipated as a result of drainage into the sand layers, Du 5 0 (at time t 5 ` ) Then the increase in effective stress in the clay layer is Dsr 5 Ds 2 Du 5 Ds 2 0 5 Ds This gradual increase in the effective stress in the clay layer will cause settlement over a period of time and is referred to as consolidation. Laboratory tests on undisturbed saturated clay specimens can be conducted (ASTM Test Designation D-2435) to determine the consolidation settlement caused by various incremental loadings. The test specimens are usually 63.5 mm in diameter and 25.4 mm in Immediately after loading: time t = 0 hi Groundwater table Sand Clay Sand Figure 1.14 Principles of consolidation 1.13 Consolidation 33 height. Specimens are placed inside a ring, with one porous stone at the top and one at the bottom of the specimen (Figure 1.15a). A load on the specimen is then applied so that the total vertical stress is equal to s. Settlement readings for the specimen are taken periodically for 24 hours. After that, the load on the specimen is doubled and more settlement readings are taken. At all times during the test, the specimen is kept under water. The procedure is continued until the desired limit of stress on the clay specimen is reached. Based on the laboratory tests, a graph can be plotted showing the variation of the void ratio e at the end of consolidation against the corresponding vertical effective stress sr. (On a semilogarithmic graph, e is plotted on the arithmetic scale and sr on the log scale.) The nature of the variation of e against log sr for a clay specimen is shown in Figure 1.15b. After the desired consolidation pressure has been reached, the specimen gradually can be unloaded, which will result in the swelling of the specimen. The figure also shows the variation of the void ratio during the unloading period. From the e–log sr curve shown in Figure 1.15b, three parameters necessary for calculating settlement in the field can be determined. They are preconsolidation pressure (scr ), compression index (Cc ), and the swelling index (Cs ). The following are more detailed descriptions for each of the parameters. Dial gauge Load Water Level Porous stone Ring Soil Specimen Porous stone (a) 2.3 c 2.2 O 2.1 D A Void ratio, e 2.0 C 1.9 (e1, 1 ) 1.8 B Slope Cc 1.7 (e3, 3 ) 1.6 Slope Cs 1.5 (e4, 4 ) (e2, 2 ) 1.4 10 100 Effective pressure, (kN/m2) (b) 400 Figure 1.15 (a) Schematic diagram of consolidation test arrangement; (b) e–log sr curve for a soft clay from East St. Louis, Illinois (Note: At the end of consolidation, s 5 sr) 34 Chapter 1: Geotechnical Properties of Soil Preconsolidation Pressure The preconsolidation pressure, scr , is the maximum past effective overburden pressure to which the soil specimen has been subjected. It can be determined by using a simple graphical procedure proposed by Casagrande (1936). The procedure involves five steps (see Figure 1.15b): a. Determine the point O on the e–log sr curve that has the sharpest curvature (i.e., the smallest radius of curvature). b. Draw a horizontal line OA. c. Draw a line OB that is tangent to the e–log sr curve at O. d. Draw a line OC that bisects the angle AOB. e. Produce the straight-line portion of the e–log sr curve backwards to intersect OC. This is point D. The pressure that corresponds to point D is the preconsolidation pressure scr . Natural soil deposits can be normally consolidated or overconsolidated (or preconsolidated). If the present effective overburden pressure sr 5 sor is equal to the preconsolidated pressure scr the soil is normally consolidated. However, if sor , scr , the soil is overconsolidated. Stas and Kulhawy (1984) correlated the preconsolidation pressure with liquidity index in the following form: scr 5 10(1.1121.62 LI) pa (1.46) where pa ⫽ atmospheric pressure (⬇100 kN/m2) LI ⫽ liquidity index A similar correlation has also been provided by Kulhawy and Mayne (1990), which is based on the work of Wood (1983) as scr 5 sor e 10c122.5LI21.25loga pa bd f sor (1.47) where o⬘ ⫽ in situ effective overburden pressure. Nagaraj and Murthy (1985) gave a correlation between c⬘ and the in situ effective overburden pressure which can be expressed as eo 1.122 2 a b 2 0.0463log sor (kN>m2 ) e L log sor (kN>m2 ) 5 0.188 where o⬘ ⫽ in situ effective overburden pressure eo ⫽ in situ void ratio LL(%) d Gs eL ⫽ void ratio at liquid limit 5 c 100 Gs ⫽ specific gravity of soil solids (1.48) 1.13 Consolidation 35 Compression Index The compression index, Cc, is the slope of the straight-line portion (the latter part) of the loading curve, or Cc 5 e1 2 e2 e1 2 e2 5 log s2r 2 log s1r s2r log ¢ ≤ s1r (1.49) where e1 and e2 are the void ratios at the end of consolidation under effective stresses s1r and s2r , respectively. The compression index, as determined from the laboratory e–log sr curve, will be somewhat different from that encountered in the field. The primary reason is that the soil remolds itself to some degree during the field exploration. The nature of variation of the e–log sr curve in the field for a normally consolidated clay is shown in Figure 1.16. The curve, generally referred to as the virgin compression curve, approximately intersects the laboratory curve at a void ratio of 0.42eo (Terzaghi and Peck, 1967). Note that eo is the void ratio of the clay in the field. Knowing the values of eo and scr , you can easily construct the virgin curve and calculate its compression index by using Eq. (1.49). The value of Cc can vary widely, depending on the soil. Skempton (1944) gave an empirical correlation for the compression index in which Cc 5 0.009(LL 2 10) (1.50) where LL 5 liquid limit . Besides Skempton, several other investigators also have proposed correlations for the compression index. Some of those are given here: Rendon-Herrero (1983): Cc 5 0.141G 1.2 s a Void ratio, e e0 0 c Virgin compression curve, Slope Cc e1 e2 1 1 eo 2.38 b Gs Laboratory consolidation curve 0.42 e0 1 2 Pressure, (log scale) Figure 1.16 Construction of virgin compression curve for normally consolidated clay (1.51) 36 Chapter 1: Geotechnical Properties of Soil Nagaraj and Murty (1985): Cc 5 0.2343 c LL(%) d Gs 100 (1.52) no 371.747 2 4.275no (1.53) Park and Koumoto (2004): Cc 5 where no 5 in situ porosity of soil. Wroth and Wood (1978): Cc 5 0.5Gs a PI(%) b 100 (1.54) If a typical value of Gs ⫽ 2.7 is used in Eq. (1.54), we obtain (Kulhawy and Mayne, 1990) Cc 5 PI(%) 74 (1.55) Swelling Index The swelling index, Cs, is the slope of the unloading portion of the e–log sr curve. In Figure 1.15b, it is defined as Cs 5 e3 2 e4 (1.56) s4r log ¢ ≤ s3r In most cases, the value of the swelling index is 41 to 15 of the compression index. Following are some representative values of Cs>Cc for natural soil deposits: Description of soil Cs , Cc Boston Blue clay Chicago clay New Orleans clay St. Lawrence clay 0.24–0.33 0.15– 0.3 0.15–0.28 0.05–0.1 The swelling index is also referred to as the recompression index. The determination of the swelling index is important in the estimation of consolidation settlement of overconsolidated clays. In the field, depending on the pressure increase, an overconsolidated clay will follow an e–log sr path abc, as shown in Figure 1.17. Note that point a, with coordinates sor and eo, corresponds to the field conditions before any increase in pressure. Point b corresponds to the preconsolidation pressure (scr ) of the clay. Line ab is approximately parallel to the laboratory unloading curve cd (Schmertmann, 1953). Hence, if you know eo, sor , scr , Cc, and Cs, you can easily construct the field consolidation curve. Using the modified Cam clay model and Eq. (1.54), Kulhawy and Mayne (1990) have shown that 1.14 Calculation of Primary Consolidation Settlement Void ratio, e e0 Slope Cs c a b Laboratory consolidation curve d Slope Cs 0.42 e0 0 Virgin compression curve, Slope Cc c Pressure, (log scale) Figure 1.17 Construction of field consolidation curve for overconsolidated clay PI(%) 370 Comparing Eqs. (1.55) and (1.57), we obtain Cs 5 1 Cs < Cc 5 1.14 37 (1.57) (1.58) Calculation of Primary Consolidation Settlement The one-dimensional primary consolidation settlement (caused by an additional load) of a clay layer (Figure 1.18) having a thickness Hc may be calculated as Sc 5 De H 1 1 eo c (1.59) Added pressure Groundwater table Sand Clay He Initial void ratio eo Sand Average effective pressure before load application o Figure 1.18 One-dimensional settlement calculation 38 Chapter 1: Geotechnical Properties of Soil where Sc 5 primary consolidation settlement De 5 total change of void ratio caused by the additional load application eo 5 void ratio of the clay before the application of load For normally consolidated clay (that is, sor 5 scr ) De 5 Cc log sor 1 Dsr sor (1.60) where sor 5 average effective vertical stress on the clay layer Dsr 5 Ds (that is, added pressure) Now, combining Eqs. (1.59) and (1.60) yields Sc 5 CcHc sor 1 Dsr log 1 1 eo sor For overconsolidated clay with sor 1 Dsr < scr , sor 1 Dsr De 5 Cs log sor (1.61) (1.62) Combining Eqs. (1.59) and (1.62) gives Sc 5 Cs Hc sor 1 Dsr log 1 1 eo sor (1.63) For overconsolidated clay, if sor , scr , sor 1 Dsr, then De 5 De1 1 De2 5 Cs log scr sor 1 Dsr 1 Cc log sor scr (1.64) Now, combining Eqs. (1.59) and (1.64) yields Sc 5 1.15 CsHc scr CcHc sor 1 Dsr log 1 log 1 1 eo sor 1 1 eo scr (1.65) Time Rate of Consolidation In Section 1.13 (see Figure 1.14), we showed that consolidation is the result of the gradual dissipation of the excess pore water pressure from a clay layer. The dissipation of pore water pressure, in turn, increases the effective stress, which induces settlement. Hence, to estimate the degree of consolidation of a clay layer at some time t after the load is applied, you need to know the rate of dissipation of the excess pore water pressure. 1.15 Time Rate of Consolidation z Ground water table h 39 u z 2H Sand Clay Hc 2H zH t t2 T T(2) t t1 T T (1) A 0 Sand (a) z0 (b) Figure 1.19 (a) Derivation of Eq. (1.68); (b) nature of variation of Du with time Figure 1.19 shows a clay layer of thickness Hc that has highly permeable sand layers at its top and bottom. Here, the excess pore water pressure at any point A at any time t after the load is applied is Du 5 (Dh)gw. For a vertical drainage condition (that is, in the direction of z only) from the clay layer, Terzaghi derived the differential equation '(Du) '2 (Du) 5 Cv 't 'z2 (1.66) where Cv 5 coefficient of consolidation, defined by Cv 5 k 5 mvgw k De g Dsr(1 1 eav ) w (1.67) in which k 5 hydraulic conductivity of the clay De 5 total change of void ratio caused by an effective stress increase of Dsr eav 5 average void ratio during consolidation mv 5 volume coefficient of compressibility 5 De>3Dsr(1 1 eav )4 Equation (1.66) can be solved to obtain Du as a function of time t with the following boundary conditions: 1. Because highly permeable sand layers are located at z 5 0 and z 5 Hc, the excess pore water pressure developed in the clay at those points will be immediately dissipated. Hence, Du 5 0 at z 5 0 and Du 5 0 at z 5 Hc 5 2H where H 5 length of maximum drainage path (due to two-way drainage condition—that is, at the top and bottom of the clay). 40 Chapter 1: Geotechnical Properties of Soil 2. At time t 5 0, Du 5 Du0 5 initial excess pore water pressure after the load is applied. With the preceding boundary conditions, Eq. (1.66) yields m5 ` 2(Du0 ) Mz 2 sin ¢ ≤ Re 2M Tv M H Du 5 a B m50 (1.68) where M 5 3(2m 1 1)p4>2 m 5 an integer 5 1, 2, c Tv 5 nondimensional time factor 5 (Cv t)>H 2 (1.69) The value of Du for various depths (i.e., z 5 0 to z 5 2H) at any given time t (and thus Tv) can be calculated from Eq. (1.68). The nature of this variation of Du is shown in Figures 1.20a and b. Figure 1.20c shows the variation of Du>Du0 with Tv and H>Hc using Eqs.(1.68) and (1.69). The average degree of consolidation of the clay layer can be defined as U5 Sc(t) (1.70) Sc(max) where Sc(t) 5 settlement of a clay layer at time t after the load is applied Sc(max) 5 maximum consolidation settlement that the clay will undergo under a given loading If the initial pore water pressure (Du0 ) distribution is constant with depth, as shown in Figure 1.20a, the average degree of consolidation also can be expressed as 2H U5 Sc(t) Sc(max) 2H 3 (Du0 )dz 2 3 ( Du)dz 5 0 0 (1.71) 2H 3 (Du0 )dz 0 or 2H (Du0 )2H 2 3 2H 0 U5 3 (Du)dz (Du0 )2H 512 (Du)dz 0 2H(Du0 ) (1.72) Now, combining Eqs. (1.68) and (1.72), we obtain U5 Sc(t) Sc(max) m5 ` 2 2 5 1 2 a ¢ 2 ≤ e 2M Tv M m50 (1.73) The variation of U with Tv can be calculated from Eq. (1.73) and is plotted in Figure 1.21. Note that Eq. (1.73) and thus Figure 1.21 are also valid when an impermeable layer is located at the bottom of the clay layer (Figure 1.20). In that case, the dissipation of 1.15 Time Rate of Consolidation Highly permeable layer (sand) Highly permeable layer (sand) u at t0 u at t0 u0 Hc 2H 41 u0 Hc H constant with depth Highly permeable layer (sand) (a) constant with depth Impermeable layer (b) 2.0 T 0 1.5 H Hc T 1 0.9 1.0 0.6 0.5 0.7 0.4 0.3 0.2 T 0.1 0.8 0.5 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Excess pore water pressure, u Initial excess pore water pressure, u0 0.9 1.0 (c) Figure 1.20 Drainage condition for consolidation: (a) two-way drainage; (b) one-way drainage; (c) plot of Du>Du0 with Tv and H>Hc excess pore water pressure can take place in one direction only. The length of the maximum drainage path is then equal to H 5 Hc. The variation of Tv with U shown in Figure 1.21 can also be approximated by Tv 5 p U% 2 ¢ ≤ 4 100 (for U 5 0 to 60%) (1.74) and Tv 5 1.781 2 0.933 log (100 2 U%) (for U . 60%) (1.75) 42 Chapter 1: Geotechnical Properties of Soil Eq (1.74) Eq (1.75) 1.0 Sand Sand Time factor, T 0.8 2H = Hc H = Hc Clay Clay 0.6 Sand u0 = constant 0.4 Rock u0 = constant 0.2 0 0 10 20 30 40 50 60 70 Average degree of consolidation, U (%) 80 90 Figure 1.21 Plot of time factor against average degree of consolidation (Du0 5 constant) Table 1.11 gives the variation of Tv with U on the basis of Eqs. (1.74) and (1.75) Sivaram and Swamee (1977) gave the following equation for U varying from 0 to 100%: (4Tv>p) 0.5 U% 5 100 31 1 (4Tv>p) 2.84 0.179 or Tv 5 (1.76) (p>4) (U%>100) 2 31 2 (U%>100) 5.64 0.357 (1.77) Equations (1.76) and (1.77) give an error in Tv of less than 1% for 0% ⬍ U ⬍ 90% and less than 3% for 90% ⬍ U ⬍ 100%. Table 1.11 Variation of Tv with U U (%) 0 1 2 3 4 5 6 7 8 9 10 Tv 0 0.00008 0.0003 0.00071 0.00126 0.00196 0.00283 0.00385 0.00502 0.00636 0.00785 U (%) Tv U (%) Tv U (%) Tv 26 27 28 29 30 31 32 33 34 35 36 0.0531 0.0572 0.0615 0.0660 0.0707 0.0754 0.0803 0.0855 0.0907 0.0962 0.102 52 53 54 55 56 57 58 59 60 61 62 0.212 0.221 0.230 0.239 0.248 0.257 0.267 0.276 0.286 0.297 0.307 78 79 80 81 82 83 84 85 86 87 88 0.529 0.547 0.567 0.588 0.610 0.633 0.658 0.684 0.712 0.742 0.774 1.15 Time Rate of Consolidation 43 Table 1.11 (Continued) U (%) 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 Tv U (%) Tv U (%) Tv U (%) Tv 0.0095 0.0113 0.0133 0.0154 0.0177 0.0201 0.0227 0.0254 0.0283 0.0314 0.0346 0.0380 0.0415 0.0452 0.0491 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 0.107 0.113 0.119 0.126 0.132 0.138 0.145 0.152 0.159 0.166 0.173 0.181 0.188 0.197 0.204 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 0.318 0.329 0.304 0.352 0.364 0.377 0.390 0.403 0.417 0.431 0.446 0.461 0.477 0.493 0.511 89 90 91 92 93 94 95 96 97 98 99 100 0.809 0.848 0.891 0.938 0.993 1.055 1.129 1.219 1.336 1.500 1.781 ⬁ Example 1.9 A laboratory consolidation test on a normally consolidated clay showed the following results: Load, Ds9 (kN , m2) Void ratio at the end of consolidation, e 140 212 0.92 0.86 The specimen tested was 25.4 mm in thickness and drained on both sides. The time required for the specimen to reach 50% consolidation was 4.5 min. A similar clay layer in the field 2.8 m thick and drained on both sides, is subjected to a similar increase in average effective pressure (i.e., s0r 5 140 kN>m2 and s0r 1 Dsr 5 212 kN>m2 ). Determine a. the expected maximum primary consolidation settlement in the field. b. the length of time required for the total settlement in the field to reach 40 mm. (Assume a uniform initial increase in excess pore water pressure with depth.) Solution Part a For normally consolidated clay [Eq. (1.49)], Cc 5 e1 2 e2 s2r log ¢ ≤ s1r 5 0.92 2 0.86 212 log ¢ ≤ 140 5 0.333 44 Chapter 1: Geotechnical Properties of Soil From Eq. (1.61), Sc 5 CcHc s0r 1 Dsr (0.333) (2.8) 212 log 5 log 5 0.0875 m 5 87.5 mm 1 1 e0 s0r 1 1 0.92 140 Part b From Eq. (1.70), the average degree of consolidation is U5 Sc(t) Sc(max) 5 40 (100) 5 45.7% 87.5 The coefficient of consolidation, Cv, can be calculated from the laboratory test. From Eq. (1.69), Tv 5 Cv t H2 For 50% consolidation (Figure 1.21), Tv 5 0.197, t 5 4.5 min, and H 5 Hc>2 5 12.7 mm, so Cv 5 T50 (0.197) (12.7) 2 H2 5 5 7.061 mm2>min t 4.5 Again, for field consolidation, U 5 45.7%. From Eq. (1.74) Tv 5 p U% 2 p 45.7 2 ¢ ≤ 5 ¢ ≤ 5 0.164 4 100 4 100 But Tv 5 Cv t H2 or 2 t5 1.16 TvH 2 5 Cv 2.8 3 1000 0.164¢ ≤ 2 7.061 5 45,523 min 5 31.6 days ■ Degree of Consolidation Under Ramp Loading The relationships derived for the average degree of consolidation in Section 1.15 assume that the surcharge load per unit area (Ds) is applied instantly at time t 5 0. However, in most practical situations, Ds increases gradually with time to a maximum value and remains constant thereafter. Figure 1.22 shows Ds increasing linearly with time (t) up to a maximum at time tc (a condition called ramp loading). For t $ tc, the magnitude of Ds 1.16 Degree of Consolidation Under Ramp Loading z 45 Sand Clay 2H = Hc Sand (a) Load per unit area, tc (b) Time, t Figure 1.22 One-dimensional consolidation due to single ramp loading remains constant. Olson (1977) considered this phenomenon and presented the average degree of consolidation, U, in the following form: For Tv # Tc, U5 Tv 2 m5 ` 1 2 b1 2 a M 4 31 2 exp(2M Tv )4 r Tc Tv m50 (1.78) and for Tv $ Tc, U512 2 m5 ` 1 2 2 a M 4 3exp(M Tc ) 2 14exp(2M Tc ) Tc m50 (1.79) where m, M, and Tv have the same definition as in Eq. (1.68) and where Tc 5 Cv tc H2 (1.80) Figure 1.23 shows the variation of U with Tv for various values of Tc, based on the solution given by Eqs. (1.78) and (1.79). Chapter 1: Geotechnical Properties of Soil 0 20 Tc 10 0.01 0.04 1 2 0.2 0.1 5 0.5 40 U (%) 46 60 80 100 0.01 0.1 1.0 10 Time factor, T Figure 1.23 Olson’s ramp-loading solution: plot of U versus Tv (Eqs. 1.78 and 1.79) Example 1.10 In Example 1.9, Part (b), if the increase in Ds would have been in the manner shown in Figure 1.24, calculate the settlement of the clay layer at time t 5 31.6 days after the beginning of the surcharge. Solution From Part (b) of Example 1.9, Cn 5 7.061 mm2>min. From Eq. (1.80), Tc 5 Cv tc H2 5 (7.061 mm2>min) (15 3 24 3 60 min) 2 2.8 3 1000 mm≤ ¢ 2 5 0.0778 72 kN/m2 tc 15 days Time, t Figure 1.24 Ramp loading 1.17 Shear Strength 47 Also, Tv 5 Cv t H2 5 (7.061 mm2>min) (31.6 3 24 3 60 min) 2 2.8 ¢ 3 1000 mm ≤ 2 5 0.164 From Figure 1.23, for Tv 5 0.164 and Tc 5 0.0778, the value of U is about 36%. Thus, Sc(t531.6 days) 5 Sc(max) (0.36) 5 (87.5) (0.36) 5 31.5 mm 1.17 ■ Shear Strength The shear strength of a soil, defined in terms of effective stress, is s 5 cr 1 sr tan fr (1.81) where sr 5 effective normal stress on plane of shearing cr 5 cohesion, or apparent cohesion fr 5 effective stress angle of friction Equation (1.81) is referred to as the Mohr–Coulomb failure criterion. The value of cr for sands and normally consolidated clays is equal to zero. For overconsolidated clays, cr . 0. For most day-to-day work, the shear strength parameters of a soil (i.e., cr and fr) are determined by two standard laboratory tests: the direct shear test and the triaxial test. Direct Shear Test Dry sand can be conveniently tested by direct shear tests. The sand is placed in a shear box that is split into two halves (Figure 1.25a). First a normal load is applied to the specimen. Then a shear force is applied to the top half of the shear box to cause failure in the sand. The normal and shear stresses at failure are sr 5 N A s5 R A and where A 5 area of the failure plane in soil—that is, the cross-sectional area of the shear box. 48 Chapter 1: Geotechnical Properties of Soil Shear stress N sc s4 tan s3 R s2 s1 1 (a) 2 3 4 (b) Effective normal stress, Figure 1.25 Direct shear test in sand: (a) schematic diagram of test equipment; (b) plot of test results to obtain the friction angle fr Several tests of this type can be conducted by varying the normal load. The angle of friction of the sand can be determined by plotting a graph of s against sr (5 s for dry sand), as shown in Figure 1.25b, or fr 5 tan 21 ¢ s ≤ sr (1.82) For sands, the angle of friction usually ranges from 26° to 45°, increasing with the relative density of compaction. A general range of the friction angle, fr, for sands is given in Table 1.12. In 1970, Brinch Hansen (see Hansbo, 1975, and Thinh, 2001) gave the following correlation for ⬘ of granular soils. ⬘ (deg) ⫽ 26° + 10Dr + 0.4Cu + 1.6 log (D50) (1.83) where Dr ⫽ relative density (fraction) Cu ⫽ uniformity coefficient D50 ⫽ mean grain size, in mm (i.e., the diameter through which 50% of the soil passes) Table 1.12 Relationship between Relative Density and Angle of Friction of Cohesionless Soils State of packing Very loose Loose Compact Dense Very dense Relative density (%) Angle of friction, f9 (deg.) ,20 20–40 40–60 60–80 .80 ,30 30–35 35–40 40–45 .45 1.17 Shear Strength 49 Teferra (1975) suggested the following empirical correlation based on a large data base. f9 (deg) 5 tan21 a 1 b ae1b (1.84) where e ⫽ void ratio a 5 2.101 1 0.097a D85 b D15 (1.85) b ⫽ 0.845 ⫺ 0.398a (1.86) D85 and D15 ⫽ diameters through which, respectively, 85% and 15% of soil passes Thinh (2001) suggested that Eq. (1.84) provides as better correlation for ⬘ compared to Eq. (1.83). Triaxial Tests Triaxial compression tests can be conducted on sands and clays Figure 1.26a shows a schematic diagram of the triaxial test arrangement. Essentially, the test consists of placing a soil specimen confined by a rubber membrane into a lucite chamber and then applying an all-around confining pressure (s3 ) to the specimen by means of the chamber fluid (generally, water or glycerin). An added stress (Ds) can also be applied to the specimen in the axial direction to cause failure ( Ds 5 Dsf at failure). Drainage from the specimen can be allowed or stopped, depending on the condition being tested. For clays, three main types of tests can be conducted with triaxial equipment (see Figure 1.27): 1. Consolidated-drained test (CD test) 2. Consolidated-undrained test (CU test) 3. Unconsolidated-undrained test (UU test) Consolidated-Drained Tests: Step 1. Apply chamber pressure s3. Allow complete drainage, so that the pore water pressure (u 5 u0 ) developed is zero. Step 2. Apply a deviator stress Ds slowly. Allow drainage, so that the pore water pressure (u 5 ud ) developed through the application of Ds is zero. At failure, Ds 5 Dsf; the total pore water pressure uf 5 u0 1 ud 5 0. So for consolidated-drained tests, at failure, Major principal effective stress 5 s3 1 Dsf 5 s1 5 s1r Minor principal effective stress 5 s3 5 s3r Changing s3 allows several tests of this type to be conducted on various clay specimens. The shear strength parameters (cr and fr) can now be determined by plotting Mohr’s circle at failure, as shown in Figure 1.26b, and drawing a common tangent to the Mohr’s circles. This is the Mohr–Coulomb failure envelope. (Note: For normally consolidated clay, cr < 0.) At failure, s1r 5 s3r tan2 ¢ 45 1 fr fr ≤ 1 2cr tan ¢45 1 ≤ 2 2 (1.87) Piston Porous stone Lucite chamber Rubber membrane Chamber fluid Soil specimen Shear stress Porous stone Base plate Valve To drainage and/or pore water pressure device Chamber fluid 1 c 3 3 1 Consolidated-drained test Schematic diagram of triaxial test equipment Effective normal stress (b) (a) Shear stress Shear stress Total stress failure envelope Effective stress failure envelope c 3 3 1 1 c Total normal stress, 3 3 1 Consolidated-undrained test (c) Shear stress Total stress failure envelope ( 0) s cu 1 3 3 1 Unconsolidated-undrained test (d) Figure 1.26 Triaxial test Normal stress (total), 1 Effective normal stress, 1.17 Shear Strength 51 3 3 3 3 3 3 3 3 Figure 1.27 Sequence of stress application in triaxial test Consolidated-Undrained Tests: Step 1. Apply chamber pressure s3. Allow complete drainage, so that the pore water pressure (u 5 u0 ) developed is zero. Step 2. Apply a deviator stress Ds. Do not allow drainage, so that the pore water pressure u 5 ud 2 0. At failure, Ds 5 Dsf; the pore water pressure uf 5 u0 1 ud 5 0 1 ud(f). Hence, at failure, Major principal total stress 5 s3 1 Dsf 5 s1 Minor principal total stress 5 s3 Major principal effective stress 5 (s3 1 Dsf ) 2 uf 5 s1r Minor principal effective stress 5 s3 2 uf 5 s3r Changing s3 permits multiple tests of this type to be conducted on several soil specimens. The total stress Mohr’s circles at failure can now be plotted, as shown in Figure 1.26c, and then a common tangent can be drawn to define the failure envelope. This total stress failure envelope is defined by the equation s 5 c 1 s tan f (1.88) where c and f are the consolidated-undrained cohesion and angle of friction, respectively. (Note: c < 0 for normally consolidated clays.) Similarly, effective stress Mohr’s circles at failure can be drawn to determine the effective stress failure envelope (Figure 1.26c), which satisfy the relation expressed in Eq. (1.81). Unconsolidated-Undrained Tests: Step 1. Apply chamber pressure s3. Do not allow drainage, so that the pore water pressure (u 5 u0 ) developed through the application of s3 is not zero. Step 2. Apply a deviator stress Ds. Do not allow drainage (u 5 ud 2 0). At failure, Ds 5 Dsf; the pore water pressure uf 5 u0 1 ud(f) For unconsolidated-undrained triaxial tests, Major principal total stress 5 s3 1 Dsf 5 s1 Minor principal total stress 5 s3 52 Chapter 1: Geotechnical Properties of Soil The total stress Mohr’s circle at failure can now be drawn, as shown in Figure 1.26d. For saturated clays, the value of s1 2 s3 5 Dsf is a constant, irrespective of the chamber confining pressure s3 (also shown in Figure 1.26d). The tangent to these Mohr’s circles will be a horizontal line, called the f 5 0 condition. The shear strength for this condition is s 5 cu 5 Dsf (1.89) 2 where cu 5 undrained cohesion (or undrained shear strength). The pore pressure developed in the soil specimen during the unconsolidatedundrained triaxial test is u 5 u0 1 ud (1.90) The pore pressure u0 is the contribution of the hydrostatic chamber pressure s3. Hence, u0 5 Bs3 (1.91) where B 5 Skempton’s pore pressure parameter. Similarly, the pore parameter ud is the result of the added axial stress Ds, so ud 5 ADs (1.92) where A 5 Skempton’s pore pressure parameter. However, Ds 5 s1 2 s3 (1.93) Combining Eqs. (1.90), (1.91), (1.92), and (1.93) gives u 5 u0 1 ud 5 Bs3 1 A(s1 2 s3 ) (1.94) The pore water pressure parameter B in soft saturated soils is approximately 1, so u 5 s3 1 A(s1 2 s3 ) (1.95) The value of the pore water pressure parameter A at failure will vary with the type of soil. Following is a general range of the values of A at failure for various types of clayey soil encountered in nature: Type of soil Sandy clays Normally consolidated clays Overconsolidated clays 1.18 A at failure 0.5–0.7 0.5–1 20.5– 0 Unconfined Compression Test The unconfined compression test (Figure 1.28a) is a special type of unconsolidatedundrained triaxial test in which the confining pressure s3 5 0, as shown in Figure 1.28b. In this test, an axial stress Ds is applied to the specimen to cause failure 1.18 Unconfined Compression Test 53 Specimen (a) Unconfined compression strength, qu Shear stress cu 3 0 1 ƒ qu Total normal stress Degree of saturation (b) (c) Figure 1.28 Unconfined compression test: (a) soil specimen; (b) Mohr’s circle for the test; (c) variation of qu with the degree of saturation (i.e., Ds 5 Dsf ). The corresponding Mohr’s circle is shown in Figure 1.28b. Note that, for this case, Major principal total stress 5 Dsf 5 qu Minor principal total stress 5 0 The axial stress at failure, Dsf 5 qu, is generally referred to as the unconfined compression strength. The shear strength of saturated clays under this condition (f 5 0), from Eq. (1.81), is s 5 cu 5 qu 2 (1.96) The unconfined compression strength can be used as an indicator of the consistency of clays. Unconfined compression tests are sometimes conducted on unsaturated soils. With the void ratio of a soil specimen remaining constant, the unconfined compression strength rapidly decreases with the degree of saturation (Figure 1.28c). 54 Chapter 1: Geotechnical Properties of Soil Comments on Friction Angle, f9 Effective Stress Friction Angle of Granular Soils In general, the direct shear test yields a higher angle of friction compared with that obtained by the triaxial test. Also, note that the failure envelope for a given soil is actually curved. The Mohr–Coulomb failure criterion defined by Eq. (1.81) is only an approximation. Because of the curved nature of the failure envelope, a soil tested at higher normal stress will yield a lower value of fr. An example of this relationship is shown in Figure 1.29, which is a plot of fr versus the void ratio e for Chattachoochee River sand near Atlanta, Georgia (Vesic, 1963). The friction angles shown were obtained from triaxial tests. Note that, for a given value of e, the magnitude of fr is about 4° to 5° smaller when the confining pressure s3r is greater than about 70 kN>m2 , compared with that when s3r , 70 kN>m2. Effective Stress Friction Angle of Cohesive Soils Figure 1.30 shows the variation of effective stress friction angle, f9, for several normally consolidated clays (Bjerrum and Simons, 1960; Kenney, 1959). It can be seen from the figure that, in general, the friction angle fr decreases with the increase in plasticity index. The value of fr generally decreases from about 37 to 38° with a plasticity index of about 10 to about 25° or less with a plasticity index of about 100. The consolidated undrained friction angle (f) of normally consolidated saturated clays generally ranges from 5 to 20°. 45 7 samples 6 samples Effective stress friction angle, (deg) 1.19 e tan = 0.68 [ 3 < 70 kN/m2] 40 8 samples 6 samples 5 samples 10 samples 7 samples 35 7 samples e tan = 0.59 [70 kN/m < 3 < 550 kN/m2 2] 30 0.6 0.7 0.8 0.9 Void ratio, e 1.0 1.1 1.2 Figure 1.29 Variation of friction angle fr with void ratio for Chattachoochee River sand (After Vesic, 1963) (From Vesic, A. B. Bearing Capacity of Deep Foundations in Sand. In Highway Research Record 39, Highway Research Board, National Research Council, Washington, D.C., 1963, Figure 11, p. 123. Reproduced with permission of the Transportation Research Board.) 1.19 Comments on Friction Angle, fr 55 1.0 Kenney (1959) Sin 0.8 Bjerrum and Simons (1960) 0.6 0.4 0.2 0 5 10 20 30 50 Plasticity index (%) 80 100 150 Figure 1.30 Variation of sin fr with plasticity index (PI) for several normally consolidated clays The consolidated drained triaxial test was described in Section 1.17. Figure 1.31 shows a schematic diagram of a plot of Ds versus axial strain in a drained triaxial test for a clay. At failure, for this test, Ds 5 Dsf. However, at large axial strain (i.e., the ultimate strength condition), we have the following relationships: Major principal stress: s1(ult) r 5 s3 1 Dsult Minor principal stress: s3(ult) r 5 s3 At failure (i.e., peak strength), the relationship between s1r and s3r is given by Eq. (1.87). However, for ultimate strength, it can be shown that s1(ult) r 5 s3r tan2 ¢ 45 1 frr ≤ 2 (1.97) where frr 5 residual effective stress friction angle. Figure 1.32 shows the general nature of the failure envelopes at peak strength and ultimate strength (or residual strength). The residual shear strength of clays is important in the evaluation of the long-term stability of new and existing slopes and the design of remedial measures. The effective stress residual friction angles frr of clays may be substantially smaller than the effective stress peak friction angle fr. Past research has shown that the clay fraction (i.e., the percent finer than 2 microns) present in a given soil, CF, and Deviator stress, f ult 3 = 3 = constant Axial strain, Figure 1.31 Plot of deviator stress versus axial strain–drained triaxial test Chapter 1: Geotechnical Properties of Soil Shear stress, c ed an dat t onsoli + c c ver s= —o h ed t ng dat stre n onsoli a t k a c Pe ally s= orm n — gth tren s tan r k s= Pea gth al stren Residu r Effective normal stress, Figure 1.32 Peak- and residual-strength envelopes for clay the clay mineralogy are the two primary factors that control frr. The following is a summary of the effects of CF on frr. 1. If CF is less than about 15%, then frr is greater than about 25°. 2. For CF . about 50%, frr is entirely governed by the sliding of clay minerals and may be in the range of about 10 to 15°. 3. For kaolinite, illite, and montmorillonite, frr is about 15°, 10°, and 5°, respectively. Illustrating these facts, Figure 1.33 shows the variation of frr with CF for several soils (Skempton, 1985). 40 Skempton (1985) pa ≈ 1 Sand Residual friction angle, r (deg) 56 30 Plasticity index, PI = 0.5 to 0.9 Clay fraction, CF 20 Kaolin 10 Bentonite 0 0 20 40 60 Clay fraction, CF (%) 80 Figure 1.33 Variation of frr with CF (Note: pa 5 atmospheric pressure) 100 1.21 Sensitivity 1.20 57 Correlations for Undrained Shear Strength, Cu Several empirical relationships can be observed between cu and the effective overburden pressure (0⬘) in the field. Some of these relationships are summarized in Table 1.13. Table 1.13 Empirical Equations Related to cu and 0⬘ Reference Relationship Remarks cu(VST) 5 0.11 1 0.00037 (PI) s0r PI ⫽ plasticity index (%) cu(VST) ⫽ undrained shear strength from vane shear test cu(VST) 5 0.11 1 0.0037 (PI) scr Skempton (1957) Chandler (1988) Jamiolkowski, et al. (1985) Mesri (1989) Bjerrum and Simons (1960) For normally consolidated clay Can be used in overconsolidated soil; accuracy ⫾25%; not valid for sensitive and fissured clays c⬘ ⫽ preconsolidation pressure cu 5 0.23 6 0.04 scr cu 5 0.22 s0r cu PI% 0.5 5 0.45a b s0r 100 For lightly overconsolidated clays Normally consolidated clay for PI ⬎ 50% cu 5 0.118 (LI) 0.15 s0r Normally consolidated clay for LI ⫽ liquidity index ⬎ 0.5 Ladd, et al. (1977) a cu b s0r overconsolidated cu a b sr0 normally consolidated 5 OCR0.8 OCR ⫽ overconsolidation ratio ⫽ c⬘/0⬘ 1.21 Sensitivity For many naturally deposited clay soils, the unconfined compression strength is much less when the soils are tested after remolding without any change in the moisture content. This property of clay soil is called sensitivity. The degree of sensitivity is the ratio of the unconfined compression strength in an undisturbed state to that in a remolded state, or St 5 qu(undisturbed) qu(remolded) (1.98) 58 Chapter 1: Geotechnical Properties of Soil The sensitivity ratio of most clays ranges from about 1 to 8; however, highly flocculent marine clay deposits may have sensitivity ratios ranging from about 10 to 80. Some clays turn to viscous liquids upon remolding, and these clays are referred to as “quick” clays. The loss of strength of clay soils from remolding is caused primarily by the destruction of the clay particle structure that was developed during the original process of sedimentation. Problems 1.1 1.2 1.3 1.4 1.5 1.6 A moist soil has a void ratio of 0.65. The moisture content of the soil is 14% and Gs ⫽ 2.7. Determine: a. Porosity b. Degree of saturation (%) c. Dry unit weight (kN/m3) For the soil described in Problem 1.1: a. What would be the saturated unit weight in kN/m3? b. How much water, in kN/m3, needs to be added to the soil for complete saturation? c. What would be the moist unit weight in kN/m3 when the degree of saturation is 70%? The moist unit weight of a soil is 18.79 kN/m3. For a moisture content of 12% and Gs ⫽ 2.65, calculate: a. Void ratio b. Porosity c. Degree of saturation d. Dry unit weight A saturated soil specimen has w ⫽ 36% and ␥d ⫽ 13.43 kN/m3. Determine: a. Void ratio b. Porosity c. Specific gravity of soil solids d. Saturated unit weight (kN/m3) The laboratory test results of a sand are emax ⫽ 0.91, emin ⫽ 0.48, and Gs ⫽ 2.67. What would be the dry and moist unit weights of this sand when compacted at a moisture content of 10% to a relative density of 65%? The laboratory test results of six soils are given in the following table. Classify the soils by the AASHTO soil classification system and give the group indices. Sieve Analysis—Percent Passing 1.7 1.8 Sieve No. Soil A Soil B Soil C Soil D Soil E Soil F 4 10 40 200 Liquid limit Plastic limit 92 48 28 13 31 26 100 60 41 33 38 25 100 98 82 72 56 31 95 90 79 64 35 26 100 91 80 30 43 29 100 82 74 55 35 21 Classify the soils in Problem 1.6 using the Unified Soil Classification System. Give group symbols and group names. The permeability of a sand was tested in the laboratory at a void ratio of 0.6 and was determined to be 0.14 cm/sec. Use Eq. (1.32) to estimate the hydraulic conductivity of this sand at a void ratio of 0.8. Problems 59 A sand has the following: D10 ⫽ 0.08 mm; D60 ⫽ 0.37 mm; void ratio e ⫽ 0.6. a. Determine the hydraulic conductivity using Eq. (1.33). b. Determine the hydraulic conductivity using Eq. (1.35). 1.10 The in situ hydraulic conductivity of a normally consolidated clay is 5.4 ⫻ 10⫺6 cm/sec at a void ratio of 0.92. What would be its hydraulic conductivity at a void ratio of 0.72? Use Eq. (1.36) and n ⫽ 5.1. 1.11 Refer to the soil profile shown in Figure P1.11. Determine the total stress, pore water pressure, and effective stress at A, B, C, and D. 1.12 For a normally consolidated clay layer, the following are given: 1.9 Thickness ⫽ 3 m Void ratio ⫽ 0.75 Liquid limit ⫽ 42 Gs ⫽ 2.72 Average effective stress on the clay layer ⫽ 110 kN/m2 How much consolidation settlement would the clay undergo if the average effective stress on the clay layer is increased to 155 kN/m2 as the result of the construction of a foundation? Use Eq. (1.51) to estimate the compression index. 1.13 Refer to Problem 1.12. Assume that the clay layer is preconsolidated, c⬘ ⫽ 130 kN/m2, and CS 5 15Cc . Estimate the consolidation settlement. 1.14 A saturated clay deposit below the ground water table in the field has liquid limit ⫽ 61%, plastic limit ⫽ 21%, and moisture content ⫽ 38%. Estimate the preconsolidation pressure, c⬘ (kN/m2) using Eq. (1.46). 1.15 A normally consolidated clay layer in the field has a thickness of 3.2 m with an average effective stress of 70 kN/m2. A laboratory consolidation test on the clay gave the following results. Pressure (kN , m2) 100 200 Void ratio 0.905 0.815 a. Determine the compression index, Cc. b. If the average effective stress on the clay layer (o⬘ + ⌬) is increased to 115 kN/m2, what would be the total consolidation settlement? A Dry sand; e = 0.55 Gs = 2.66 3m B 1.5 m Water table Sand Gs = 2.66 e = 0.48 C 5m Clay w = 34.78% Gs = 2.74 D Rock Figure P1.11 60 Chapter 1: Geotechnical Properties of Soil 1.16 A clay soil specimen, 25.4 mm thick (drained on top and bottom), was tested in the laboratory. For a given load increment, the time for 50% consolidation was 5 min 20 sec. How long will it take for 50% consolidation of a similar clay layer in the field that is 2.5 m thick and drained on one side only? 1.17 A clay soil specimen 25.4 mm thick (drained on top only) was tested in the laboratory. For a given load increment, the time for 60% consolidation was 6 min 20 sec. How long will it take for 50% consolidation for a similar clay layer in the field that is 3.05 m thick and drained on both sides? 1.18 Refer to Figure P1.18. A total of 60-mm consolidation settlement is expected in the two clay layers due to a surcharge ⌬. Find the duration of surcharge application at which 30 mm of total settlement would take place. 1.19 The coefficient of consolidation of a clay for a given pressure range was obtained as 8 ⫻ 10⫺3 mm2/sec on the basis of one-dimensional consolidation test results. In the field, there is a 2-m thick layer of the same clay (Figure P1.19a). Based on the assumption that a uniform surcharge of 70 kN/m2 was to be applied instantaneously, the total consolidation settlement was estimated to be 150 mm. However, during construction, the loading was gradual; the resulting surcharge can be approximated as shown in Figure P1.19b. Estimate the settlement at t ⫽ 30 days and t ⫽ 120 days after the beginning of construction. 1.20 A direct shear test was conducted on a 2 ⫻ 2 specimen of dry sand had the following results. Normal force (N) 146.8 245.4 294.3 Shear force at failure (N) 91.9 159.2 178.8 Draw a graph of shear stress at failure versus normal stress and determine the soil friction angle. 1m Groundwater table 1m Sand 2m Clay C = 2 mm2/min 1m Sand 1m Clay C = 2 mm2/min Sand Figure P1.18 Problems 61 (kN/m2) Sand 1m Clay 2m 70 kN/m2 Sand 60 (a) Time, days (b) Figure P1.19 1.21 For a sand, given: D85 ⫽ 0.21 mm D50 ⫽ 0.13 mm D15 ⫽ 0.09 mm Uniformity coefficient, Cu ⫽ 2.1 Void ratio, e ⫽ 0.68 Relative density ⫽ 53% Estimate the soil friction angle using a. Eq. (1.83) b. Eq. (1.84) 1.22 A consolidated-drained triaxial test on a normally consolidated clay yields the following results. All around confining pressure, 3⬘ ⫽ 138 kN/m2 Added axial stress at failure, ⌬ ⫽ 276 kN/m2 Determine the shear-strength parameters. 1.23 The following are the results of two consolidated-drained triaxial tests on a clay. Test I: 3⬘ ⫽ 82.8 kN/m2; 1⬘(failure) ⫽ 329.2 kN/m2 Test II: 3⬘ ⫽ 165.6 kN/m2; 1⬘(failure) ⫽ 558.6 kN/m2 Determine the shear-strength parameters—that is, c⬘ and ⬘. 1.24 A consolidated-undrained triaxial test was conducted on a saturated normally consolidated clay. The test results are 3 ⫽ 89.7 kN/m2 1(failure) ⫽ 220.8 kN/m2 Pore water pressure at failure ⫽ 37.95 kN/m2 Determine c, , c⬘, and ⬘. 1.25 A normally consolidated clay soil has ⫽ 20° and ⬘ ⫽ 28°. If a consolidatedundrained test is conducted on this clay with an all around pressure of 3 ⫽ 148.35 kN/m2, what would be the magnitude of the major principal stress, 1, and the pore water pressure, u, at failure? 62 Chapter 1: Geotechnical Properties of Soil References AMER, A. M., and AWAD, A. A. (1974). “Permeability of Cohesionless Soils,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 100, No. GT12, pp. 1309–1316. AMERICAN SOCIETY FOR TESTING AND MATERIALS (2009). Annual Book of ASTM Standards, Vol. 04.08, West Conshohocken, PA. BJERRUM, L., and SIMONS, N. E. (1960). “Comparison of Shear Strength Characteristics of Normally Consolidated Clay,” Proceedings, Research Conference on Shear Strength of Cohesive Soils, ASCE, pp. 711–726. CARRIER III, W. D. (2003). “Goodbye, Hazen; Hello, Kozeny-Carman,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 129, No. 11, pp. 1054–1056. CASAGRANDE, A. (1936). “Determination of the Preconsolidation Load and Its Practical Significance,” Proceedings, First International Conference on Soil Mechanics and Foundation Engineering, Cambridge, MA, Vol. 3, pp. 60–64. CHANDLER, R. J. (1988). “The in situ Measurement of the Undrained Shear Strength of Clays Using the Field Vane,” STP 1014, Vane Shear Strength Testing in Soils: Field and Laboratory Studies, American Society for Testing and Materials, pp. 13–44. CHAPUIS, R. P. (2004). “Predicting the Saturated Hydraulic Conductivity of Sand and Gravel Using Effective Diameter and Void Ratio,” Canadian Geotechnical Journal, Vol. 41, No. 5, pp. 787–795. DARCY, H. (1856). Les Fontaines Publiques de la Ville de Dijon, Paris. DAS, B. M. (2009). Soil Mechanics Laboratory Manual, 7th ed., Oxford University Press, New York. HANSBO, S. (1975). Jordmateriallära: 211, Stockholm, Awe/Gebers. HIGHWAY RESEARCH BOARD (1945). Report of the Committee on Classification of Materials for Subgrades and Granular Type Roads, Vol. 25, pp. 375– 388. JAMILKOWSKI, M., LADD, C. C., GERMAINE, J. T., and LANCELLOTTA, R. (1985). “New Developments in Field and Laboratory Testing of Soils,” Proceedings, XI International Conference on Soil Mechanics and Foundations Engineering, San Francisco, Vol. 1, pp. 57–153. KENNEY, T. C. (1959). “Discussion,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 85, No. SM3, pp. 67–69. KULHAWY, F. H., and MAYNE, P. W. (1990). Manual of Estimating Soil Properties for Foundation Design, Electric Power Research Institute, Palo Alto, California. LADD, C. C., FOOTE, R., ISHIHARA, K., SCHLOSSER, F., and POULOS, H. G. (1977). “Stress Deformation and Strength Characteristics,” Proceedings, Ninth International Conference on Soil Mechanics and Foundation Engineering, Tokyo, Vol. 2, pp. 421–494. MESRI, G. (1989). “A Re-evaluation of su(mob) ⬇ 0.22p Using Laboratory Shear Tests,” Canadian Geotechnical Journal, Vol. 26, No. 1, pp. 162–164. MESRI, G., and OLSON, R. E. (1971). “Mechanism Controlling the Permeability of Clays,” Clay and Clay Minerals, Vol. 19, pp. 151–158. NAGARAJ, T. S., and MURTHY, B. R. S. (1985). “Prediction of the Preconsolidation Pressure and Recompression Index of Soils,” Geotechnical Testing Journal, American Society for Testing and Materials, Vol. 8, No. 4, pp. 199–202. OLSON, R. E. (1977). “Consolidation Under Time-Dependent Loading,” Journal of Geotechnical Engineering, ASCE, Vol. 103, No. GT1, pp. 55–60. PARK, J. H., and KOUMOTO, T. (2004). “New Compression Index Equation,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 130, No. 2, pp. 223–226. RENDON-HERRERO, O. (1980). “Universal Compression Index Equation,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 106, No. GT11, pp. 1178–1200. SAMARASINGHE, A. M., HUANG, Y. H., and DRNEVICH, V. P. (1982). “Permeability and Consolidation of Normally Consolidated Soils,” Journal of the Geotechnical Engineering Division, ASCE, Vol. 108, No. GT6, pp. 835 –850. References 63 SCHMERTMANN, J. H. (1953). “Undisturbed Consolidation Behavior of Clay,” Transactions, American Society of Civil Engineers, Vol. 120, p. 1201. SIVARAM, B., and SWAMEE, A. (1977), “A Computational Method for Consolidation Coefficient,” Soils and Foundations, Vol. 17, No. 2, pp. 48–52. SKEMPTON, A. W. (1944). “Notes on the Compressibility of Clays,” Quarterly Journal of Geological Society, London, Vol. C, pp. 119–135. SKEMPTON, A. W. (1953). “The Colloidal Activity of Clays,” Proceedings, 3rd International Conference on Soil Mechanics and Foundation Engineering, London, Vol. 1, pp. 57–61. SKEMPTON, A. W. (1957). “The Planning and Design of New Hong Kong Airport,” Proceedings, The Institute of Civil Engineers, London, Vol. 7, pp. 305 –307. SKEMPTON, A. W. (1985). “Residual Strength of Clays in Landslides, Folded Strata, and the Laboratory,” Geotechnique, Vol. 35, No. 1, pp. 3–18. STAS, C. V., and KULHAWY, F. H. (1984). “Critical Evaluation of Design Methods for Foundations Under Axial Uplift and Compression Loading, REPORT EL-3771, Electric Power Research Institute, Palo Alto, California. TAVENAS, F., JEAN, P., LEBLOND, F. T. P., and LEROUEIL, S. (1983). “The Permeability of Natural Soft Clays. Part II: Permeability Characteristics,” Canadian Geotechnical Journal, Vol. 20, No. 4, pp. 645–660. TAYLOR, D. W. (1948). Fundamentals of Soil Mechanics, Wiley, New York. TEFERRA, A. (1975). Beziehungen zwischen Reibungswinkel, Lagerungsdichte und Sonderwiderständen nichtbindiger Böden mit verschiedener Kornverteilung. Ph.D. Thesis, Technical University of Aachen Germany. TERZAGHI, K., and PECK, R. B. (1967). Soil Mechanics in Engineering Practice, Wiley, New York. THINH, K. D. (2001). “How Reliable is Your Angle of Internal Friction?” Proceedings, XV International Conference on Soil Mechanics and Geotechnical Engineering, Istanbul, Turkey, Vol. 1, pp. 73–76. VESIC, A. S. (1963). “Bearing Capacity of Deep Foundations in Sand,” Highway Research Record No. 39, National Academy of Sciences, Washington, DC., pp. 112–154. WOOD, D. M. (1983). “Index Properties and Critical State Soil Mechanics,” Proceedings, Symposium on Recent Developments in Laboratory and Field Tests and Analysis of Geotechnical Problems, Bangkok, p. 309. WROTH, C. P., and WOOD, D. M. (1978). “The Correlation of Index Properties with Some Basic Engineering Properties of Soils,” Canadian Geotechnical Journal, Vol. 15, No. 2, pp. 137–145. 2 2.1 Natural Soil Deposits and Subsoil Exploration Introduction To design a foundation that will support a structure, an engineer must understand the types of soil deposits that will support the foundation. Moreover, foundation engineers must remember that soil at any site frequently is nonhomogeneous; that is, the soil profile may vary. Soil mechanics theories involve idealized conditions, so the application of the theories to foundation engineering problems involves a judicious evaluation of site conditions and soil parameters. To do this requires some knowledge of the geological process by which the soil deposit at the site was formed, supplemented by subsurface exploration. Good professional judgment constitutes an essential part of geotechnical engineering—and it comes only with practice. This chapter is divided into two parts. The first is a general overview of natural soil deposits generally encountered, and the second describes the general principles of subsoil exploration. Natural Soil Deposits 2.2 Soil Origin Most of the soils that cover the earth are formed by the weathering of various rocks. There are two general types of weathering: (1) mechanical weathering and (2) chemical weathering. Mechanical weathering is a process by which rocks are broken down into smaller and smaller pieces by physical forces without any change in the chemical composition. Changes in temperature result in expansion and contraction of rock due to gain and loss of heat. Continuous expansion and contraction will result in the development of cracks in rocks. Flakes and large fragments of rocks are split. Frost action is another source of mechanical weathering of rocks. Water can enter the pores, cracks, and other openings in the rock. When the temperature drops, the water freezes, thereby increasing the volume by about 9%. This results in an outward pressure from inside the rock. Continuous freezing and thawing will result in the breakup of a rock mass. Exfoliation is another mechanical 64 2.2 Soil Origin 65 weathering process by which rock plates are peeled off from large rocks by physical forces. Mechanical weathering of rocks also takes place due to the action of running water, glaciers, wind, ocean waves, and so forth. Chemical weathering is a process of decomposition or mineral alteration in which the original minerals are changed into something entirely different. For example, the common minerals in igneous rocks are quartz, feldspars, and ferromagnesian minerals. The decomposed products of these minerals due to chemical weathering are listed in Table 2.1. Table 2.1 Some Decomposed Products of Minerals in Igneous Rock Mineral Decomposed Product Quartz Quartz (sand grains) Potassium feldspar (KAlSi3O8) and Sodium feldspar (NaAlSi3O8) Kaolinite (clay) Bauxite Illite (clay) Silica Calcium feldspar (CaAl2Si2O8) Silica Calcite Biotite Clay Limonite Hematite Silica Calcite Olivine (Mg, Fe)2SiO4 Limonite Serpentine Hematite Silica Most rock weathering is a combination of mechanical and chemical weathering. Soil produced by the weathering of rocks can be transported by physical processes to other places. The resulting soil deposits are called transported soils. In contrast, some soils stay where they were formed and cover the rock surface from which they derive. These soils are referred to as residual soils. Transported soils can be subdivided into five major categories based on the transporting agent: 1. 2. 3. 4. 5. Gravity transported soil Lacustrine (lake) deposits Alluvial or fluvial soil deposited by running water Glacial deposited by glaciers Aeolian deposited by the wind In addition to transported and residual soils, there are peats and organic soils, which derive from the decomposition of organic materials. 66 Chapter 2: Natural Soil Deposits and Subsoil Exploration Residual Soil Residual soils are found in areas where the rate of weathering is more than the rate at which the weathered materials are carried away by transporting agents. The rate of weathering is higher in warm and humid regions compared to cooler and drier regions and, depending on the climatic conditions, the effect of weathering may vary widely. Residual soil deposits are common in the tropics, on islands such as the Hawaiian Islands, and in the southeastern United States. The nature of a residual soil deposit will generally depend on the parent rock. When hard rocks such as granite and gneiss undergo weathering, most of the materials are likely to remain in place. These soil deposits generally have a top layer of clayey or silty clay material, below which are silty or sandy soil layers. These layers in turn are generally underlain by a partially weathered rock and then sound bedrock. The depth of the sound bedrock may vary widely, even within a distance of a few meters. Figure 2.1 shows the boring log of a residual soil deposit derived from the weathering of granite. In contrast to hard rocks, there are some chemical rocks, such as limestone, that are chiefly made up of calcite (CaCO3 ) mineral. Chalk and dolomite have large concentrations of dolomite minerals 3Ca Mg(CO3 ) 24. These rocks have large amounts of soluble materials, Fines (percent passing No. 200 sieve) 0 20 40 60 80 100 0 Light brown silty clay (Unified Soil Classification— MH) Light brown clayey silt (Unified Soil Classification— MH) Silty sand (Unified Soil Classification— SC to SP) 1 2 3 Depth (m) 2.3 4 5 6 Partially decomposed granite 7 8 Bedrock 9 Figure 2.1 Boring log for a residual soil derived from granite 2.4 Gravity Transported Soil 67 some of which are removed by groundwater, leaving behind the insoluble fraction of the rock. Residual soils that derive from chemical rocks do not possess a gradual transition zone to the bedrock, as seen in Figure 2.1. The residual soils derived from the weathering of limestone-like rocks are mostly red in color. Although uniform in kind, the depth of weathering may vary greatly. The residual soils immediately above the bedrock may be normally consolidated. Large foundations with heavy loads may be susceptible to large consolidation settlements on these soils. 2.4 Gravity Transported Soil Residual soils on a natural slope can move downwards. Cruden and Varnes (1996) proposed a velocity scale for soil movement on a slope, which is summarized in Table 2.2. When residual soils move down a natural slope very slowly, the process is usually referred to as creep. When the downward movement of soil is sudden and rapid, it is called a landslide. The deposits formed by down-slope creep and landslides are colluvium. Table 2.2 Velocity Scale for Soil Movement on a Slope Description Very slow Slow Moderate Rapid Velocity (mm/sec) 5 ⫻ 10⫺5 to 5 ⫻ 10⫺7 5 ⫻ 10⫺3 to 5 ⫻ 10⫺5 5 ⫻ 10⫺1 to 5 ⫻ 10⫺3 5 ⫻ 101 to 5 ⫻ 10⫺1 Colluvium is a heterogeneous mixture of soils and rock fragments ranging from clay-sized particles to rocks having diameters of one meter or more. Mudflows are one type of gravity-transported soil. Flows are downward movements of earth that resemble a viscous fluid (Figure 2.2) and come to rest in a more dense condition. The soil deposits derived from past mudflows are highly heterogeneous in composition. Mud flow Figure 2.2 Mudflow 68 Chapter 2: Natural Soil Deposits and Subsoil Exploration 2.5 Alluvial Deposits Alluvial soil deposits derive from the action of streams and rivers and can be divided into two major categories: (1) braided-stream deposits and (2) deposits caused by the meandering belt of streams. Deposits from Braided Streams Braided streams are high-gradient, rapidly flowing streams that are highly erosive and carry large amounts of sediment. Because of the high bed load, a minor change in the velocity of flow will cause sediments to deposit. By this process, these streams may build up a complex tangle of converging and diverging channels separated by sandbars and islands. The deposits formed from braided streams are highly irregular in stratification and have a wide range of grain sizes. Figure 2.3 shows a cross section of such a deposit. These deposits share several characteristics: 1. The grain sizes usually range from gravel to silt. Clay-sized particles are generally not found in deposits from braided streams. 2. Although grain size varies widely, the soil in a given pocket or lens is rather uniform. 3. At any given depth, the void ratio and unit weight may vary over a wide range within a lateral distance of only a few meters. This variation can be observed during soil exploration for the construction of a foundation for a structure. The standard penetration resistance at a given depth obtained from various boreholes will be highly irregular and variable. Alluvial deposits are present in several parts of the western United States, such as Southern California, Utah, and the basin and range sections of Nevada. Also, a large amount of sediment originally derived from the Rocky Mountain range was carried eastward to form the alluvial deposits of the Great Plains. On a smaller scale, this type of natural soil deposit, left by braided streams, can be encountered locally. Meander Belt Deposits The term meander is derived from the Greek word maiandros, after the Maiandros (now Menderes) River in Asia, famous for its winding course. Mature streams in a valley curve back and forth. The valley floor in which a river meanders is referred to as the meander Fine sand Gravel Silt Coarse sand Figure 2.3 Cross section of a braided-stream deposit 2.5 Alluvial Deposits 69 Erosion Deposition (point bar) Deposition (point bar) River Figure 2.4 Formation of point bar deposits and oxbow lake in a meandering stream Oxbow lake Erosion belt. In a meandering river, the soil from the bank is continually eroded from the points where it is concave in shape and is deposited at points where the bank is convex in shape, as shown in Figure 2.4. These deposits are called point bar deposits, and they usually consist of sand and silt-size particles. Sometimes, during the process of erosion and deposition, the river abandons a meander and cuts a shorter path. The abandoned meander, when filled with water, is called an oxbow lake. (See Figure 2.4.) During floods, rivers overflow low-lying areas. The sand and silt-size particles carried by the river are deposited along the banks to form ridges known as natural levees (Figure 2.5). Levee deposit Clay plug Backswamp deposit Lake River Figure 2.5 Levee and backswamp deposit 70 Chapter 2: Natural Soil Deposits and Subsoil Exploration Table 2.3 Properties of Deposits within the Mississippi Alluvial Valley Environment Soil texture Natural water content (%) Liquid limit Plasticity index Natural levee Clay (CL) Silt (ML) 25–35 15–35 35–45 NP–35 15–25 NP–5 Point bar Silt (ML) and silty sand (SM) 25–45 30–55 10–25 Abandoned channel Clay (CL, CH) 30–95 30–100 10–65 Backswamps Clay (CH) 25–70 40–115 25–100 Swamp Organic clay (OH) 100–265 135–300 100–165 (Note: NP—Nonplastic) Finer soil particles consisting of silts and clays are carried by the water farther onto the floodplains. These particles settle at different rates to form what is referred to as backswamp deposits (Figure 2.5), often highly plastic clays. Table 2.3 gives some properties of soil deposits found in natural levees, point bars, abandoned channels, backswamps and swamps within the alluvial Mississippi Valley (Kolb and Shockley, 1959). 2.6 Lacustrine Deposits Water from rivers and springs flows into lakes. In arid regions, streams carry large amounts of suspended solids. Where the stream enters the lake, granular particles are deposited in the area forming a delta. Some coarser particles and the finer particles (that is, silt and clay) that are carried into the lake are deposited onto the lake bottom in alternate layers of coarse-grained and fine-grained particles. The deltas formed in humid regions usually have finer grained soil deposits compared to those in arid regions. Varved clays are alternate layers of silt and silty clay with layer thicknesses rarely exceeding about 13 mm. The silt and silty clay that constitute the layers were carried into fresh water lakes by melt water at the end of the Ice Age. The hydraulic conductivity (Section 1.10) of varved clays exhibits a high degree of anisotropy. 2.7 Glacial Deposits During the Pleistocene Ice Age, glaciers covered large areas of the earth. The glaciers advanced and retreated with time. During their advance, the glaciers carried large amounts of sand, silt, clay, gravel, and boulders. Drift is a general term usually applied to the deposits laid down by glaciers. The drifts can be broadly divided into two major categories: (a) unstratified drifts and (b) stratified drifts. A brief description of each category follows. 2.8 Aeolian Soil Deposits 71 Unstratified Drifts The unstratified drifts laid down by melting glaciers are referred to as till. The physical characteristics of till may vary from glacier to glacier. Till is called clay till because of the presence of the large amount of clay-sized particles in it. In some areas, tills constitute large amounts of boulders, and they are referred to as boulder till. The range of grain sizes in a given till varies greatly. The amount of clay-sized fractions present and the plasticity indices of tills also vary widely. During the field exploration program, erratic values of standard penetration resistance (Section 2.13) also may be expected. The land forms that developed from the till deposits are called moraines. A terminal moraine (Figure 2.6) is a ridge of till that marks the maximum limit of a glacier’s advance. Recessional moraines are ridges of till developed behind the terminal moraine at varying distances apart. They are the result of temporary stabilization of the glacier during the recessional period. The till deposited by the glacier between the moraines is referred to as ground moraine (Figure 2.6). Ground moraines constitute large areas of the central United States and are called till plains. Terminal moraine Outwash Ground moraine Outwash plain Figure 2.6 Terminal moraine, ground moraine, and outwash plain Stratified Drifts The sand, silt, and gravel that are carried by the melting water from the front of a glacier are called outwash. The melted water sorts out the particles by the grain size and forms stratified deposits. In a pattern similar to that of braided-stream deposits, the melted water also deposits the outwash, forming outwash plains (Figure 2.6), also called glaciofluvial deposits. 2.8 Aeolian Soil Deposits Wind is also a major transporting agent leading to the formation of soil deposits. When large areas of sand lie exposed, wind can blow the sand away and redeposit it elsewhere. Deposits of windblown sand generally take the shape of dunes (Figure 2.7). As dunes are formed, the sand is blown over the crest by the wind. Beyond the crest, the sand particles roll down the slope. The process tends to form a compact sand deposit on the windward side, and a rather loose deposit on the leeward side, of the dune. Dunes exist along the southern and eastern shores of Lake Michigan, the Atlantic Coast, the southern coast of California, and at various places along the coasts of Oregon and 72 Chapter 2: Natural Soil Deposits and Subsoil Exploration Sand particle Wind direction Figure 2.7 Sand dune Washington. Sand dunes can also be found in the alluvial and rocky plains of the western United States. Following are some of the typical properties of dune sand: 1. The grain-size distribution of the sand at any particular location is surprisingly uniform. This uniformity can be attributed to the sorting action of the wind. 2. The general grain size decreases with distance from the source, because the wind carries the small particles farther than the large ones. 3. The relative density of sand deposited on the windward side of dunes may be as high as 50 to 65%, decreasing to about 0 to 15% on the leeward side. Figure 2.8 shows a sand dune from the Thar Desert, which is a large and arid region located in the northwestern part of India that covers an area of about 200,000 square kilometers. Loess is an aeolian deposit consisting of silt and silt-sized particles. The grain-size distribution of loess is rather uniform. The cohesion of loess is generally derived from a clay Figure 2.8 A sand dune from the Thar Desert, India (Courtesy of A. S. Wayal, K. J. Somaiya Polytechnic, Mumbai, India) 2.10 Some Local Terms for Soils 73 coating over the silt-sized particles, which contributes to a stable soil structure in an unsaturated state. The cohesion may also be the result of the precipitation of chemicals leached by rainwater. Loess is a collapsing soil, because when the soil becomes saturated, it loses its binding strength between particles. Special precautions need to be taken for the construction of foundations over loessial deposits. There are extensive deposits of loess in the United States, mostly in the midwestern states of Iowa, Missouri, Illinois, and Nebraska and for some distance along the Mississippi River in Tennessee and Mississippi. Volcanic ash (with grain sizes between 0.25 to 4 mm) and volcanic dust (with grain sizes less than 0.25 mm) may be classified as wind-transported soil. Volcanic ash is a lightweight sand or sandy gravel. Decomposition of volcanic ash results in highly plastic and compressible clays. 2.9 Organic Soil Organic soils are usually found in low-lying areas where the water table is near or above the ground surface. The presence of a high water table helps in the growth of aquatic plants that, when decomposed, form organic soil. This type of soil deposit is usually encountered in coastal areas and in glaciated regions. Organic soils show the following characteristics: 1. Their natural moisture content may range from 200 to 300%. 2. They are highly compressible. 3. Laboratory tests have shown that, under loads, a large amount of settlement is derived from secondary consolidation. 2.10 Some Local Terms for Soils Soils are sometimes referred to by local terms. The following are a few of these terms with a brief description of each. 1. Caliche: a Spanish word derived from the Latin word calix, meaning lime. It is mostly found in the desert southwest of the United States. It is a mixture of sand, silt, and gravel bonded together by calcareous deposits. The calcareous deposits are brought to the surface by a net upward migration of water. The water evaporates in the high local temperature. Because of the sparse rainfall, the carbonates are not washed out of the top layer of soil. 2. Gumbo: a highly plastic, clayey soil. 3. Adobe: a highly plastic, clayey soil found in the southwestern United States. 4. Terra Rossa: residual soil deposits that are red in color and derive from limestone and dolomite. 5. Muck: organic soil with a very high moisture content. 6. Muskeg: organic soil deposit. 7. Saprolite: residual soil deposit derived from mostly insoluble rock. 8. Loam: a mixture of soil grains of various sizes, such as sand, silt, and clay. 9. Laterite: characterized by the accumulation of iron oxide (Fe2 O3) and aluminum oxide (Al2 O3) near the surface, and the leaching of silica. Lateritic soils in Central America contain about 80 to 90% of clay and silt-size particles. In the United States, lateritic soils can be found in the southeastern states, such as Alabama, Georgia, and the Carolinas. 74 Chapter 2: Natural Soil Deposits and Subsoil Exploration Subsurface Exploration 2.11 Purpose of Subsurface Exploration The process of identifying the layers of deposits that underlie a proposed structure and their physical characteristics is generally referred to as subsurface exploration. The purpose of subsurface exploration is to obtain information that will aid the geotechnical engineer in 1. 2. 3. 4. Selecting the type and depth of foundation suitable for a given structure. Evaluating the load-bearing capacity of the foundation. Estimating the probable settlement of a structure. Determining potential foundation problems (e.g., expansive soil, collapsible soil, sanitary landfill, and so on). 5. Determining the location of the water table. 6. Predicting the lateral earth pressure for structures such as retaining walls, sheet pile bulkheads, and braced cuts. 7. Establishing construction methods for changing subsoil conditions. Subsurface exploration may also be necessary when additions and alterations to existing structures are contemplated. 2.12 Subsurface Exploration Program Subsurface exploration comprises several steps, including the collection of preliminary information, reconnaissance, and site investigation. Collection of Preliminary Information This step involves obtaining information regarding the type of structure to be built and its general use. For the construction of buildings, the approximate column loads and their spacing and the local building-code and basement requirements should be known. The construction of bridges requires determining the lengths of their spans and the loading on piers and abutments. A general idea of the topography and the type of soil to be encountered near and around the proposed site can be obtained from the following sources: 1. United States Geological Survey maps. 2. State government geological survey maps. 3. United States Department of Agriculture’s Soil Conservation Service county soil reports. 4. Agronomy maps published by the agriculture departments of various states. 5. Hydrological information published by the United States Corps of Engineers, including records of stream flow, information on high flood levels, tidal records, and so on. 6. Highway department soil manuals published by several states. The information collected from these sources can be extremely helpful in planning a site investigation. In some cases, substantial savings may be realized by anticipating problems that may be encountered later in the exploration program. 2.12 Subsurface Exploration Program 75 Reconnaissance The engineer should always make a visual inspection of the site to obtain information about 1. The general topography of the site, the possible existence of drainage ditches, abandoned dumps of debris, and other materials present at the site. Also, evidence of creep of slopes and deep, wide shrinkage cracks at regularly spaced intervals may be indicative of expansive soils. 2. Soil stratification from deep cuts, such as those made for the construction of nearby highways and railroads. 3. The type of vegetation at the site, which may indicate the nature of the soil. For example, a mesquite cover in central Texas may indicate the existence of expansive clays that can cause foundation problems. 4. High-water marks on nearby buildings and bridge abutments. 5. Groundwater levels, which can be determined by checking nearby wells. 6. The types of construction nearby and the existence of any cracks in walls or other problems. The nature of the stratification and physical properties of the soil nearby also can be obtained from any available soil-exploration reports on existing structures. Site Investigation The site investigation phase of the exploration program consists of planning, making test boreholes, and collecting soil samples at desired intervals for subsequent observation and laboratory tests. The approximate required minimum depth of the borings should be predetermined. The depth can be changed during the drilling operation, depending on the subsoil encountered. To determine the approximate minimum depth of boring, engineers may use the rules established by the American Society of Civil Engineers (1972): 1. Determine the net increase in the effective stress, Ds r, under a foundation with depth as shown in Figure 2.9. (The general equations for estimating increases in stress are given in Chapter 5.) 2. Estimate the variation of the vertical effective stress, sor , with depth. D σ σo Figure 2.9 Determination of the minimum depth of boring 76 Chapter 2: Natural Soil Deposits and Subsoil Exploration 3. Determine the depth, D 5 D1, at which the effective stress increase Dsr is equal to ( 101 )q (q 5 estimated net stress on the foundation). 4. Determine the depth, D 5 D2, at which Dsr>sor 5 0.05. 5. Choose the smaller of the two depths, D1 and D2, just determined as the approximate minimum depth of boring required, unless bedrock is encountered. If the preceding rules are used, the depths of boring for a building with a width of 30 m will be approximately the following, according to Sowers and Sowers (1970): No. of stories Boring depth 1 2 3 4 5 3.5 m 6m 10 m 16 m 24 m To determine the boring depth for hospitals and office buildings, Sowers and Sowers (1970) also used the following rules. • For light steel or narrow concrete buildings, Db S0.7 5a (2.1) where Db ⫽ depth of boring S ⫽ number of stories a 5 3 if Db is in meters • For heavy steel or wide concrete buildings, Db S0.7 5b (2.2) where b5 6 if Db is in meters 20 if Db is in feet When deep excavations are anticipated, the depth of boring should be at least 1.5 times the depth of excavation. Sometimes, subsoil conditions require that the foundation load be transmitted to bedrock. The minimum depth of core boring into the bedrock is about 3 m. If the bedrock is irregular or weathered, the core borings may have to be deeper. There are no hard-and-fast rules for borehole spacing. Table 2.4 gives some general guidelines. Spacing can be increased or decreased, depending on the condition of the subsoil. If various soil strata are more or less uniform and predictable, fewer boreholes are needed than in nonhomogeneous soil strata. 2.13 Exploratory Borings in the Field 77 Table 2.4 Approximate Spacing of Boreholes Type of project Spacing (m) Multistory building One-story industrial plants Highways Residential subdivision Dams and dikes 10–30 20–60 250–500 250–500 40–80 The engineer should also take into account the ultimate cost of the structure when making decisions regarding the extent of field exploration. The exploration cost generally should be 0.1 to 0.5% of the cost of the structure. Soil borings can be made by several methods, including auger boring, wash boring, percussion drilling, and rotary drilling. 2.13 Exploratory Borings in the Field Auger boring is the simplest method of making exploratory boreholes. Figure 2.10 shows two types of hand auger: the posthole auger and the helical auger. Hand augers cannot be used for advancing holes to depths exceeding 3 to 5 m. However, they can be used for soil exploration work on some highways and small structures. Portable power-driven helical augers (76 mm to 305 mm in diameter) are available for making deeper boreholes. The soil samples obtained from such borings are highly disturbed. In some noncohesive soils or soils having low cohesion, the walls of the boreholes will not stand unsupported. In such circumstances, a metal pipe is used as a casing to prevent the soil from caving in. (a) (b) Figure 2.10 Hand tools: (a) posthole auger; (b) helical auger 78 Chapter 2: Natural Soil Deposits and Subsoil Exploration When power is available, continuous-flight augers are probably the most common method used for advancing a borehole. The power for drilling is delivered by truck- or tractor-mounted drilling rigs. Boreholes up to about 60 to 70 m can easily be made by this method. Continuous-flight augers are available in sections of about 1 to 2 m with either a solid or hollow stem. Some of the commonly used solid-stem augers have outside diameters of 66.68 mm, 82.55 mm, 101.6 mm, and 114.3 mm. Common commercially available hollow-stem augers have dimensions of 63.5 mm ID and 158.75 mm OD, 69.85 mm ID and 177.8 OD, 76.2 mm ID and 203.2 OD, and 82.55 mm ID and 228.6 mm OD. The tip of the auger is attached to a cutter head (Figure 2.11). During the drilling operation (Figure 2.12), section after section of auger can be added and the hole extended downward. The flights of the augers bring the loose soil from the bottom of the hole to the surface. The driller can detect changes in the type of soil by noting changes in the speed and sound of drilling. When solid-stem augers are used, the auger must be withdrawn at regular intervals to obtain soil samples and also to conduct other operations such as standard penetration tests. Hollow-stem augers have a distinct advantage over solid-stem augers in that they do not have to be removed frequently for Figure 2.11 Carbide-tipped cutting head on auger flight (Courtesy of Braja M. Das, Henderson, NV) 2.13 Exploratory Borings in the Field 79 Figure 2.12 Drilling with continuous-flight augers (Danny R. Anderson, PE of Professional Service Industries, Inc, El Paso, Texas.) sampling or other tests. As shown schematically in Figure 2.13, the outside of the hollowstem auger acts as a casing. The hollow-stem auger system includes the following components: Outer component: Inner component: (a) hollow auger sections, (b) hollow auger cap, and (c) drive cap (a) pilot assembly, (b) center rod column, and (c) rod-to-cap adapter The auger head contains replaceable carbide teeth. During drilling, if soil samples are to be collected at a certain depth, the pilot assembly and the center rod are removed. The soil sampler is then inserted through the hollow stem of the auger column. Wash boring is another method of advancing boreholes. In this method, a casing about 2 to 3 m long is driven into the ground. The soil inside the casing is then removed by means of a chopping bit attached to a drilling rod. Water is forced through the drilling rod and exits at a very high velocity through the holes at the bottom of the chopping bit (Figure 2.14). The water and the chopped soil particles rise in the drill hole and overflow at the top of the casing through a T connection. The washwater is 80 Chapter 2: Natural Soil Deposits and Subsoil Exploration Rope Drive cap Rod-to-cap adapter Auger connector Hollow-stem auger section Derrick Pressure water Tub Center rod Casing Drill rod Pilot assembly Auger connector Chopping bit Auger head Center head Replaceable carbide auger tooth Figure 2.13 Hollow-stem auger components (After ASTM, 2001) (ASTM D4700-91: Standard Guide for Soil Sampling from the Vadose Zone. Copyright ASTM INTERNATIONAL. Reprinted with permission.) Driving shoe Water jet at high velocity Figure 2.14 Wash boring collected in a container. The casing can be extended with additional pieces as the borehole progresses; however, that is not required if the borehole will stay open and not cave in. Wash borings are rarely used now in the United States and other developed countries. Rotary drilling is a procedure by which rapidly rotating drilling bits attached to the bottom of drilling rods cut and grind the soil and advance the borehole. There are several types of drilling bit. Rotary drilling can be used in sand, clay, and rocks (unless they are badly fissured). Water or drilling mud is forced down the drilling rods to the bits, and the return flow forces the cuttings to the surface. Boreholes with diameters of 50 to 203 mm can easily be made by this technique. The drilling mud is a slurry of water and bentonite. Generally, it is used when the soil that is encountered is likely to cave in. When soil samples are needed, the drilling rod is raised and the drilling bit is replaced by a sampler. With the environmental drilling applications, rotary drilling with air is becoming more common. Percussion drilling is an alternative method of advancing a borehole, particularly through hard soil and rock. A heavy drilling bit is raised and lowered to chop the hard soil. The chopped soil particles are brought up by the circulation of water. Percussion drilling may require casing. 2.15 Split-Spoon Sampling 2.14 81 Procedures for Sampling Soil Two types of soil samples can be obtained during subsurface exploration: disturbed and undisturbed. Disturbed, but representative, samples can generally be used for the following types of laboratory test: 1. 2. 3. 4. 5. Grain-size analysis Determination of liquid and plastic limits Specific gravity of soil solids Determination of organic content Classification of soil Disturbed soil samples, however, cannot be used for consolidation, hydraulic conductivity, or shear strength tests. Undisturbed soil samples must be obtained for these types of laboratory tests. Sections 2.15 through 2.18 describe some procedures for obtaining soil samples during field exploration. 2.15 Split-Spoon Sampling Split-spoon samplers can be used in the field to obtain soil samples that are generally disturbed, but still representative. A section of a standard split-spoon sampler is shown in Figure 2.15a. The tool consists of a steel driving shoe, a steel tube that is split longitudinally Water port Head 457.2 mm (18 in.) Pin 76.2 mm (3 in.) 50.8 mm 34.93 mm (2 in.) in.) (1-3/8 Ball valve Drilling rod Coupling Threads Driving shoe Split barrel (a) (b) Figure 2.15 (a) Standard split-spoon sampler; (b) spring core catcher 50.8 mm (2 in.) 82 Chapter 2: Natural Soil Deposits and Subsoil Exploration in half, and a coupling at the top. The coupling connects the sampler to the drill rod. The standard split tube has an inside diameter of 34.93 mm and an outside diameter of 50.8 mm; however, samplers having inside and outside diameters up to 63.5 mm and 76.2 mm, respectively, are also available. When a borehole is extended to a predetermined depth, the drill tools are removed and the sampler is lowered to the bottom of the hole. The sampler is driven into the soil by hammer blows to the top of the drill rod. The standard weight of the hammer is 622.72 N, and for each blow, the hammer drops a distance of 0.762 m. The number of blows required for a spoon penetration of three 152.4-mm intervals are recorded. The number of blows required for the last two intervals are added to give the standard penetration number, N, at that depth. This number is generally referred to as the N value (American Society for Testing and Materials, 2001, Designation D-1586-99). The sampler is then withdrawn, and the shoe and coupling are removed. Finally, the soil sample recovered from the tube is placed in a glass bottle and transported to the laboratory. This field test is called the standard penetration test (SPT). Figure 2.16a and b show a split-spoon sampler unassembled before and after sampling. The degree of disturbance for a soil sample is usually expressed as AR (%) 5 D2o 2 D2i D2i (100) (2.3) where AR 5 area ratio (ratio of disturbed area to total area of soil) Do 5 outside diameter of the sampling tube Di 5 inside diameter of the sampling tube When the area ratio is 10% or less, the sample generally is considered to be undisturbed. For a standard split-spoon sampler, A R (%) 5 (50.8) 2 2 (34.93) 2 (34.93) 2 (100) 5 111.5% Figure 2.16 (a) Unassembled split-spoon sampler; (b) after sampling (Courtesy of Professional Service Industries, Inc. (PSI), Waukesha, Wisconsin) 2.15 Split-Spoon Sampling 83 Hence, these samples are highly disturbed. Split-spoon samples generally are taken at intervals of about 1.5 m. When the material encountered in the field is sand (particularly fine sand below the water table), recovery of the sample by a split-spoon sampler may be difficult. In that case, a device such as a spring core catcher may have to be placed inside the split spoon (Figure 2.15b). At this juncture, it is important to point out that several factors contribute to the variation of the standard penetration number N at a given depth for similar soil profiles. Among these factors are the SPT hammer efficiency, borehole diameter, sampling method, and rod length (Skempton, 1986; Seed, et al., 1985). The SPT hammer energy efficiency can be expressed as Er (%) 5 actual hammer energy to the sampler 3 100 input energy Theoretical input energy 5 Wh (2.4) (2.5) where W 5 weight of the hammer < 0.623 kN h 5 height of drop < 0.76 mm So, Wh 5 (0.623) (0.76) 5 0.474 kN-m In the field, the magnitude of Er can vary from 30 to 90%. The standard practice now in the U.S. is to express the N-value to an average energy ratio of 60% (<N60 ). Thus, correcting for field procedures and on the basis of field observations, it appears reasonable to standardize the field penetration number as a function of the input driving energy and its dissipation around the sampler into the surrounding soil, or N60 5 NhH hB hS hR 60 (2.6) where N60 5 standard penetration number, corrected for field conditions N 5 measured penetration number hH 5 hammer efficiency (%) hB 5 correction for borehole diameter hS 5 sampler correction hR 5 correction for rod length Variations of hH, hB, hS, and hR, based on recommendations by Seed et al. (1985) and Skempton (1986), are summarized in Table 2.5. Correlations for N60 in Cohesive Soil Besides compelling the geotechnical engineer to obtain soil samples, standard penetration tests provide several useful correlations. For example, the consistency of clay soils can be estimated from the standard penetration number, N60. In order to achieve that, Szechy and Vargi (1978) calculated the consistency index (CI) as 84 Chapter 2: Natural Soil Deposits and Subsoil Exploration Table 2.5 Variations of hH, hB, hS, and hR [Eq. (2.6)] 1. Variation of hH 2. Variation of hB Country Hammer type Hammer release hH (%) Japan Donut Donut Safety Donut Donut Donut Donut Free fall Rope and pulley Rope and pulley Rope and pulley Rope and pulley Free fall Rope and pulley 78 67 60 45 45 60 50 United States Argentina China hS Standard sampler With liner for dense sand and clay With liner for loose sand 1.0 0.8 0.9 CI 5 hB 60 –120 150 200 1 1.05 1.15 4. Variation of hR 3. Variation of hS Variable Diameter mm Rod length m hR .10 6 –10 4–6 0– 4 1.0 0.95 0.85 0.75 LL 2 w LL 2 PL (2.7) where w 5 natural moisture content LL 5 liquid limit PL 5 plastic limit The approximate correlation between CI, N60, and the unconfined compression strength (qu) is given in Table 2.6. Hara, et al. (1971) also suggested the following correlation between the undrained shear strength of clay (cu) and N60: cu 5 0.29N 0.72 60 pa (2.8) where pa 5 atmospheric pressure (< 100 kN>m2; < 2000 lb>in2 ). Table 2.6 Approximate Correlation between CI, N60, and qu Standard penetration number, N60 ,2 2–8 8–15 15–30 .30 Consistency Very soft Soft to medium Stiff Very stiff Hard CI Unconfined compression strength, qu (kN/m2) ,0.5 0.5–0.75 0.75–1.0 1.0–1.5 .1.5 ,25 25–80 80–150 150–400 .400 2.15 Split-Spoon Sampling 85 The overconsolidation ratio, OCR, of a natural clay deposit can also be correlated with the standard penetration number. On the basis of the regression analysis of 110 data points, Mayne and Kemper (1988) obtained the relationship OCR 5 0.193¢ N60 0.689 ≤ sor (2.9) where sor 5 effective vertical stress in MN>m2 . It is important to point out that any correlation between cu, OCR, and N60 is only approximate. Correction for N60 in Granular Soil In granular soils, the value of N is affected by the effective overburden pressure, sor . For that reason, the value of N60 obtained from field exploration under different effective overburden pressures should be changed to correspond to a standard value of sor . That is, (N1 ) 60 5 CNN60 (2.10) where (N1 ) 60 5 value of N60 corrected to a standard value of sor 3100 kN>m2 A2000 lb>ft 2 B4 CN 5 correction factor N60 5 value of N obtained from field exploration [Eq. (2.6)] In the past, a number of empirical relations were proposed for CN. Some of the relationships are given next. The most commonly cited relationships are those of Liao and Whitman (1986) and Skempton (1986). In the following relationships for CN, note that sor is the effective overburden pressure and pa 5 atmospheric pressure A< 100 kN>m2 B Liao and Whitman’s relationship (1986): CN 5 1 0.5 C sor S ¢ ≤ pa (2.11) Skempton’s relationship (1986): CN 5 2 11 ¢ CN 5 sor ≤ pa 3 21 ¢ sor ≤ pa (for normally consolidated fine sand) (2.12) (for normally consolidated coarse sand) (2.13) 86 Chapter 2: Natural Soil Deposits and Subsoil Exploration CN 5 1.7 sor 0.7 1 ¢ ≤ pa (for overconsolidated sand) (2.14) Seed et al.’s relationship (1975): CN 5 1 21.25 log¢ sor ≤ pa (2.15) Peck et al.’s relationship (1974): CN 5 0.77 log 20 £ sor § ¢ ≤ pa ¢for sor $ 0.25≤ pa (2.16) Bazaraa (1967): CN 5 CN 5 4 sor 1 1 4a b pa afor 4 3.25 1 a sor b pa sor # 0.75b pa afor (2.17) sor . 0.75b pa (2.18) Table 2.7 shows the comparison of CN derived using various relationships cited above. It can be seen that the magnitude of the correction factor estimated by using any one of the relationships is approximately the same, considering the uncertainties involved in conducting the standard penetration tests. Hence, it is recommended that Eq. (2.11) may be used for all calculations. Table 2.7 Variation of CN CN so9 pa 0.25 0.50 0.75 1.00 1.50 2.00 3.00 4.00 Eq. (2.11) Eq. (2.12) 2.00 1.41 1.15 1.00 0.82 0.71 0.58 0.50 1.60 1.33 1.14 1.00 0.80 0.67 0.50 0.40 Eq. (2.13) 1.33 1.20 1.09 1.00 0.86 0.75 0.60 0.60 Eq. (2.14) Eq. (2.15) Eq. (2.16) Eqs. (2.17) and (2.18) 1.78 1.17 1.17 1.00 0.77 0.63 0.46 0.36 1.75 1.38 1.15 1.00 0.78 0.62 0.40 0.25 1.47 1.23 1.10 1.00 0.87 0.77 0.63 0.54 2.00 1.33 1.00 0.94 0.84 0.76 0.65 0.55 2.15 Split-Spoon Sampling 87 Correlation between N60 and Relative Density of Granular Soil An approximate relationship between the corrected standard penetration number and the relative density of sand is given in Table 2.8. The values are approximate primarily because the effective overburden pressure and the stress history of the soil significantly influence the N60 values of sand. Kulhawy and Mayne (1990) modified an empirical relationship for relative density that was given by Marcuson and Bieganousky (1977), which can be expressed as Dr (%) 5 12.2 1 0.75B222N60 1 2311 2 711OCR 2 779¢ 0.5 sor (2.19) ≤ 2 50C2u R pa where Dr ⫽ relative density o⬘ ⫽ effective overburden pressure Cu ⫽ uniformity coefficient of sand preconsolidation pressure, scr OCR 5 effective overburden pressure, sor pa 5 atmospheric pressure Meyerhof (1957) developed a correlation between Dr and N60 as N60 5 c17 1 24a sor b d D2r pa or 0.5 Dr 5 μ N60 s9o c17 1 24a b d pa ∂ (2.20) Equation (2.20) provides a reasonable estimate only for clean, medium fine sand. Cubrinovski and Ishihara (1999) also proposed a correlation between N60 and the relative density of sand (Dr ) that can be expressed as N60 ¢0.23 1 Dr (%) 5 D 9 0.06 1.7 ≤ D50 0.5 1 T (100) s £ or ≥ pa Table 2.8 Relation between the Corrected (N1 )60 Values and the Relative Density in Sands Standard penetration number, (N1)60 Approximate relative density, Dr (%) 0–5 5–10 10–30 30–50 0–5 5–30 30–60 60–95 (2.21) 88 Chapter 2: Natural Soil Deposits and Subsoil Exploration where pa 5 atmospheric pressure (< 100 kN>m2 ) D50 5 sieve size through which 50% of the soil will pass (mm) Kulhawy and Mayne (1990) correlated the corrected standard penetration number and the relative density of sand in the form 0.5 (N1 ) 60 Dr (%) 5 C S CpCACOCR (100) (2.22) where CP 5 grain-size correlations factor 5 60 1 25 logD50 (2.23) t b 100 COCR 5 correlation factor for overconsolidation 5 OCR0.18 D50 5 diameter through which 50% soil will pass through (mm) t 5 age of soil since deposition (years) OCR 5 overconsolidation ratio CA 5 correlation factor for aging 5 1.2 1 0.05 loga (2.24) (2.25) Correlation between Angle of Friction and Standard Penetration Number The peak friction angle, fr, of granular soil has also been correlated with N60 or (N1 ) 60 by several investigators. Some of these correlations are as follows: 1. Peck, Hanson, and Thornburn (1974) give a correlation between N60 and fr in a graphical form, which can be approximated as (Wolff, 1989) fr (deg) 5 27.1 1 0.3N60 2 0.000543N604 2 (2.26) 2. Schmertmann (1975) provided the correlation between N60, sor , and fr. Mathematically, the correlation can be approximated as (Kulhawy and Mayne, 1990) fr 5 tan21 £ N60 12.2 1 20.3 a where N60 5 field standard penetration number sor 5 effective overburden pressure pa 5 atmospheric pressure in the same unit as sor fr 5 soil friction angle 0.34 sor § b pa (2.27) 2.16 Sampling with a Scraper Bucket 89 3. Hatanaka and Uchida (1996) provided a simple correlation between fr and (N1 ) 60 that can be expressed as fr 5 "20(N1 ) 60 1 20 (2.28) The following qualifications should be noted when standard penetration resistance values are used in the preceding correlations to estimate soil parameters: 1. The equations are approximate. 2. Because the soil is not homogeneous, the values of N60 obtained from a given borehole vary widely. 3. In soil deposits that contain large boulders and gravel, standard penetration numbers may be erratic and unreliable. Although approximate, with correct interpretation the standard penetration test provides a good evaluation of soil properties. The primary sources of error in standard penetration tests are inadequate cleaning of the borehole, careless measurement of the blow count, eccentric hammer strikes on the drill rod, and inadequate maintenance of water head in the borehole. Correlation between Modulus of Elasticity and Standard Penetration Number The modulus of elasticity of granular soils (Es) is an important parameter in estimating the elastic settlement of foundations. A first order estimation for Es was given by Kulhawy and Mayne (1990) as Es 5 aN60 pa (2.29) where pa 5 atmospheric pressure (same unit as Es ) 5 for sands with fines a 5 • 10 for clean normally consolidated sand 15 for clean overconsolidated sand 2.16 Sampling with a Scraper Bucket When the soil deposits are sand mixed with pebbles, obtaining samples by split spoon with a spring core catcher may not be possible because the pebbles may prevent the springs from closing. In such cases, a scraper bucket may be used to obtain disturbed representative samples (Figure 2.17). The scraper bucket has a driving point and can be attached to a drilling rod. The sampler is driven down into the soil and rotated, and the scrapings from the side fall into the bucket. 90 Chapter 2: Natural Soil Deposits and Subsoil Exploration S Drill rod S Driving point Section at S – S Figure 2.17 Scraper bucket 2.17 Sampling with a Thin-Walled Tube Thin-walled tubes are sometimes referred to as Shelby tubes. They are made of seamless steel and are frequently used to obtain undisturbed clayey soils. The most common thin-walled tube samplers have outside diameters of 50.8 mm and 76.2 mm. The bottom end of the tube is sharpened. The tubes can be attached to drill rods (Figure 2.18). The drill rod with the sampler attached is lowered to the bottom of the borehole, and the sampler is pushed into the soil. The soil sample inside the tube is then pulled out. The two ends are sealed, and the sampler is sent to the laboratory for testing. Figure 2.19 shows the sequence of sampling with a thin-walled tube in the field. Samples obtained in this manner may be used for consolidation or shear tests. A thin-walled tube with a 50.8-mm (2-in.) outside diameter has an inside diameter of about 47.63 mm (178 in.). The area ratio is A R (%) 5 D2o 2 D2i D2i (100) 5 (50.8) 2 2 (47.63) 2 (47.63) 2 (100) 5 13.75% Increasing the diameters of samples increases the cost of obtaining them. Drill rod Figure 2.18 Thin-walled tube Thin-walled tube 2.17 Sampling with a Thin-Walled Tube 91 (a) (b) Figure 2.19 Sampling with a thin-walled tube: (a) tube being attached to drill rod; (b) tube sampler pushed into soil (Courtesy of Khaled Sobhan, Florida Atlantic Univetsity, Boca Raton, Florida) 92 Chapter 2: Natural Soil Deposits and Subsoil Exploration (c) Figure 2.19 (continued) (c) recovery of soil sample (Courtesy of Khaled Sobhan, Florida Atlantic University, Boca Raton, Florida) 2.18 Sampling with a Piston Sampler When undisturbed soil samples are very soft or larger than 76.2 mm in diameter, they tend to fall out of the sampler. Piston samplers are particularly useful under such conditions. There are several types of piston sampler; however, the sampler proposed by Osterberg (1952) is the most useful. (see Figures 2.20a and 2.20b). It consists of a thin-walled tube with a piston. Initially, the piston closes the end of the tube. The sampler is lowered to the bottom of the borehole (Figure 2.20a), and the tube is pushed into the soil hydraulically, past the piston. Then the pressure is released through a hole in the piston rod (Figure 2.20b). To a large extent, the presence of the piston prevents distortion in the sample by not letting the soil squeeze into the sampling tube very fast and by not admitting excess soil. Consequently, samples obtained in this manner are less disturbed than those obtained by Shelby tubes. 2.19 Observation of Water Tables The presence of a water table near a foundation significantly affects the foundation’s loadbearing capacity and settlement, among other things. The water level will change seasonally. In many cases, establishing the highest and lowest possible levels of water during the life of a project may become necessary. 2.19 Observation of Water Tables 93 Water (in) Drill rod Water (out) Vent Piston (a) Sample (b) Figure 2.20 Piston sampler: (a) sampler at the bottom of borehole; (b) tube pushed into the soil hydraulically If water is encountered in a borehole during a field exploration, that fact should be recorded. In soils with high hydraulic conductivity, the level of water in a borehole will stabilize about 24 hours after completion of the boring. The depth of the water table can then be recorded by lowering a chain or tape into the borehole. In highly impermeable layers, the water level in a borehole may not stabilize for several weeks. In such cases, if accurate water-level measurements are required, a piezometer can be used. A piezometer basically consists of a porous stone or a perforated pipe with a plastic standpipe attached to it. Figure 2.21 shows the general placement of a piezometer in a borehole. This procedure will allow periodic checking until the water level stabilizes. 94 Chapter 2: Natural Soil Deposits and Subsoil Exploration Protective cover Piezometer water level Groundwater level Standpipe Bentonite cement grout Bentonite plug Filter tip Sand 2.20 Figure 2.21 Casagrande-type piezometer (Courtesy of N. Sivakugan, James Cook University, Australia.) Vane Shear Test The vane shear test (ASTM D-2573) may be used during the drilling operation to determine the in situ undrained shear strength (cu ) of clay soils—particularly soft clays. The vane shear apparatus consists of four blades on the end of a rod, as shown in Figure 2.22. The height, H, of the vane is twice the diameter, D. The vane can be either rectangular or tapered (see Figure 2.22). The dimensions of vanes used in the field are given in Table 2.9. The vanes of the apparatus are pushed into the soil at the bottom of a borehole without disturbing the soil appreciably. Torque is applied at the top of the rod to rotate the vanes at a standard rate of 0.1°>sec. This rotation will induce failure in a soil of cylindrical shape surrounding the vanes. The maximum torque, T, applied to cause failure is measured. Note that T 5 f(cu, H, and D) (2.30) 95 L = 10D 2.20 Vane Shear Test H = 2D 45 D D Rectangular vane Tapered vane Figure 2.22 Geometry of field vane (After ASTM, 2001) (Annual Book of ASTM Standards, Vol. 04.08. Copyright ASTM INTERNATIONAL. Reprinted with permission.) or cu 5 T K (2.31) where T is in N # m, cu is in kN>m2, and K 5 a constant with a magnitude depending on the dimension and shape of the vane The constant K5 ¢ p D2H D ≤ ¢ ≤ ¢1 1 ≤ 6 2 3H 10 (2.32a) 96 Chapter 2: Natural Soil Deposits and Subsoil Exploration Table 2.9 ASTM Recommended Dimensions of Field Vanesa (Annual Book of ASTM Standards, Vol. 04.08. Copyright ASTM INTERNATIONAL. Reprinted with permission.) Casing size AX BX NX 101.6 mmb Diameter, D mm Height, H mm Thickness of blade mm Diameter of rod mm 38.1 50.8 63.5 92.1 76.2 101.6 127.0 184.1 1.6 1.6 3.2 3.2 12.7 12.7 12.7 12.7 a The selection of a vane size is directly related to the consistency of the soil being tested; that is, the softer the soil, the larger the vane diameter should be. b Inside diameter. where D 5 diameter of vane in cm H 5 measured height of vane in cm If H>D 5 2, Eq. (2.32a) yields K 5 366 3 1028D3 c (cm) (2.32b) In English units, if cu and T in Eq. (2.31) are expressed in lb>ft 2 and lb-ft, respectively, then K5 ¢ p D2H D ≤¢ ≤ ¢1 1 ≤ 1728 2 3H (2.33a) If H>D 5 2, Eq. (2.33a) yields K 5 0.0021D3 c (in.) (2.33b) Field vane shear tests are moderately rapid and economical and are used extensively in field soil-exploration programs. The test gives good results in soft and medium-stiff clays and gives excellent results in determining the properties of sensitive clays. Sources of significant error in the field vane shear test are poor calibration of torque measurement and damaged vanes. Other errors may be introduced if the rate of rotation of the vane is not properly controlled. For actual design purposes, the undrained shear strength values obtained from field vane shear tests 3cu(VST) 4 are too high, and it is recommended that they be corrected according to the equation cu(corrected) 5 lcu(VST) (2.34) where l 5 correction factor. 2.20 Vane Shear Test 97 Several correlations have been given previously for the correction factor l. The most commonly used correlation for l is that given by Bjerrum (1972), which can be expressed as l 5 1.7 2 0.54 log 3PI(%)4 (2.35a) Morris and Williams (1994) provided the following correlations: l 5 1.18e 20.08(PI) 1 0.57 (for PI . 5) l 5 7.01e 20.08(LL) 1 0.57 (where LL is in %) (2.35b) (2.35c) The field vane shear strength can be correlated with the preconsolidation pressure and the overconsolidation ratio of the clay. Using 343 data points, Mayne and Mitchell (1988) derived the following empirical relationship for estimating the preconsolidation pressure of a natural clay deposit: scr 5 7.043cu(field) 4 0.83 (2.36) Here, scr 5 preconsolidation pressure (kN>m2 ) cu(field) 5 field vane shear strength (kN>m2 ) The overconsolidation ratio, OCR, also can be correlated to cu(field) according to the equation OCR 5 b cu(field) sor (2.37) where sor 5 effective overburden pressure. The magnitudes of b developed by various investigators are given below. • Mayne and Mitchell (1988): b 5 223PI(%)4 20.48 • Hansbo (1957): 222 w(%) (2.39) 1 0.08 1 0.0055(PI) (2.40) b5 • (2.38) Larsson (1980): b5 98 Chapter 2: Natural Soil Deposits and Subsoil Exploration 2.21 Cone Penetration Test The cone penetration test (CPT), originally known as the Dutch cone penetration test, is a versatile sounding method that can be used to determine the materials in a soil profile and estimate their engineering properties. The test is also called the static penetration test, and no boreholes are necessary to perform it. In the original version, a 60° cone with a base area of 10 cm2 was pushed into the ground at a steady rate of about 20 mm>sec, and the resistance to penetration (called the point resistance) was measured. The cone penetrometers in use at present measure (a) the cone resistance (qc ) to penetration developed by the cone, which is equal to the vertical force applied to the cone, divided by its horizontally projected area; and (b) the frictional resistance (fc ), which is the resistance measured by a sleeve located above the cone with the local soil surrounding it. The frictional resistance is equal to the vertical force applied to the sleeve, divided by its surface area—actually, the sum of friction and adhesion. Generally, two types of penetrometers are used to measure qc and fc: 1. Mechanical friction-cone penetrometer (Figure 2.23). The tip of this penetrometer is connected to an inner set of rods. The tip is first advanced about 40 mm, giving the 35.7 mm 15 mm 15 mm 12.5 mm 30 mm dia. 47 mm 52.5 mm 45 mm 187 mm 11.5 mm 133.5 mm 20 mm dia. 35.7 mm 25 mm 387 mm 266 mm 69 mm 23 mm dia. 33.5 mm 32.5 mm dia. 146 mm 35.7 mm dia. 30 mm 35 mm 60 Collapsed Extended Figure 2.23 Mechanical friction-cone penetrometer (After ASTM, 2001) (Annual Book of ASTM Standards, Vol. 04.08. Copyright ASTM INTERNATIONAL. Reprinted with permission.) 2.21 Cone Penetration Test 99 cone resistance. With further thrusting, the tip engages the friction sleeve. As the inner rod advances, the rod force is equal to the sum of the vertical force on the cone and sleeve. Subtracting the force on the cone gives the side resistance. 2. Electric friction-cone penetrometer (Figure 2.24). The tip of this penetrometer is attached to a string of steel rods. The tip is pushed into the ground at the rate of 20 mm>sec. Wires from the transducers are threaded through the center of the rods and continuously measure the cone and side resistances. Figure 2.25 shows a photograph of an electric friction-cone penetrometer. 7 8 6 5 3 4 3 2 1 35.6 mm 1 2 3 4 5 6 7 8 Conical point (10 cm2) Load cell Strain gauges Friction sleeve (150 cm2) Adjustment ring Waterproof bushing Cable Connection with rods Figure 2.24 Electric friction-cone penetrometer (After ASTM, 2001) (Annual Book of ASTM Standards, Vol. 04.08. Copyright ASTM INTERNATIONAL. Reprinted with permission.) Figure 2.25 Photograph of an electric friction-cone penetrometer (Courtesy of Sanjeev Kumar, Southern Illinois University, Carbondale, Illinois) 100 Chapter 2: Natural Soil Deposits and Subsoil Exploration Figure 2.26 shows the sequence of a cone penetration test in the field. A truckmounted CPT rig is shown in Figure 2.26a. A hydraulic ram located inside the truck pushes the cone into the ground. Figure 2.26b shows the cone penetrometer in the truck being put in the proper location. Figure 2.26c shows the progress of the CPT. (a) Figure 2.26 Cone penetration test in field: (a) mounted CPT rig; (b) cone penetrometer being set in proper location (Courtesy of Sanjeev Kumar, Southern Illinois University, Carbondale, Illinois) (a) 2.21 Cone Penetration Test 101 Figure 2.26 (continued) (c) test in progress (Courtesy of Sanjeev Kumar, Southern Illinois University, Carbondale, Illinois) Figure 2.27 shows the results of penetrometer test in a soil profile with friction measurement by an electric friction-cone penetrometer. Several correlations that are useful in estimating the properties of soils encountered during an exploration program have been developed for the point resistance (qc ) and the friction ratio (Fr ) obtained from the cone penetration tests. The friction ratio is defined as Fr 5 fc frictional resistance 5 qc cone resistance (2.41) In a more recent study on several soils in Greece, Anagnostopoulos et al. (2003) expressed Fr as Fr (%) 5 1.45 2 1.36 logD50 (electric cone) (2.42) Fr (%) 5 0.7811 2 1.611 logD50 (mechanical cone) (2.43) and where D50 5 size through which 50% of soil will pass through (mm). The D50 for soils based on which Eqs. (2.42) and (2.43) have been developed ranged from 0.001 mm to about 10 mm. As in the case of standard penetration tests, several correlations have been developed between qc and other soil properties. Some of these correlations are presented next. 102 Chapter 2: Natural Soil Deposits and Subsoil Exploration qc (kN/m2) 5,000 0 10,000 0 0 2 2 4 4 6 Depth (m) Depth (m) 0 fc (kN/m2) 200 400 6 8 8 10 10 12 12 Figure 2.27 Cone penetrometer test with friction measurement Correlation between Relative Density (Dr ) and qc for Sand Lancellotta (1983) and Jamiolkowski et al. (1985) showed that the relative density of normally consolidated sand, Dr, and qc can be correlated according to the formula (Figure 2.28). Dr (%) 5 A 1 B log 10 ¢ qc "sor ≤ (2.44) The preceding relationship can be rewritten as (Kulhawy and Mayne, 1990) Dr (%) 5 68Clog £ qc "pa ? s0r ≥ 2 1S (2.45) 2.21 Cone Penetration Test 103 95 85 Dr = –98 + 66 log10 qc (σ 0 )0.5 75 Dr (%) 65 2σ 55 qc and σ 0 in ton (metric)/m2 2σ 45 Ticino sand Ottawa sand Edgar sand 35 Hokksund sand 25 Hilton mine sand 15 1000 100 qc σ 0 0.5 Figure 2.28 Relationship between Dr and qc (Based on Lancellotta, 1983, and Jamiolski et al., 1985) where pa 5 atmospheric pressure (< 100 kN>m2 ) sor 5 vertical effective stress Baldi et al. (1982), and Robertson and Campanella (1983) recommended the empirical relationship shown in Figure 2.29 between vertical effective stress (sor ), relative density (Dr ), and qc for normally consolidated sand. Kulhawy and Mayne (1990) proposed the following relationship to correlate Dr, qc, and the vertical effective stress sor : Dr 5 qc pa 1 R≥ ¥ 305QcOCR1.8 sor 0.5 ¢ ≤ ï pa B In this equation, OCR 5 overconsolidation ratio pa 5 atmospheric pressure Qc 5 compressibility factor (2.46) 104 Chapter 2: Natural Soil Deposits and Subsoil Exploration 0 Cone point resistance, qc (MN/m2) 10 20 30 40 50 0 Vertical effective stress, σo (kN/m2) 100 200 300 400 500 Dr = 40% 50% 60% 70% 80% 90% Figure 2.29 Variation of qc, sor , and Dr for normally consolidated quartz sand (Based on Baldi et al., 1982, and Robertson and Campanella, 1983) The recommended values of Qc are as follows: Highly compressible sand 5 0.91 Moderately compressible sand 5 1.0 Low compressible sand 5 1.09 Correlation between qc and Drained Friction Angle (f9) for Sand On the basis of experimental results, Robertson and Campanella (1983) suggested the variation of Dr, sor , and fr for normally consolidated quartz sand. This relationship can be expressed as (Kulhawy and Mayne, 1990) fr 5 tan21 B0.1 1 0.38 log¢ qc ≤R sor (2.47) Based on the cone penetration tests on the soils in the Venice Lagoon (Italy), Ricceri et al. (2002) proposed a similar relationship for soil with classifications of ML and SP-SM as fr 5 tan21 B0.38 1 0.27 log¢ qc ≤R sor (2.48) In a more recent study, Lee et al. (2004) developed a correlation between fr, qc , and the horizontal effective stress (shr ) in the form fr 5 15.575¢ qc 0.1714 ≤ shr (2.49) 2.21 Cone Penetration Test Clay 1000 Clayey silt & Sandy silt silty clay and silt Silty Sand 105 Sand 900 qc (kN/m2) N60 700 Ratio, 800 400 600 500 Range of results of Robertson & Campanella (1983) 300 200 Average of Robertson & Campanella (1983) 100 0 0.001 0.01 0.1 Mean grain size, D50 (mm) 1.0 Figure 2.30 General range of variation of qc>N60 for various types of soil Correlation between qc and N60 Figure 2.30 shows a plot of qc (kN> m2)> N60 (N60 5 standard penetration resistance) against the mean grain size (D50 in mm) for various types of soil. This was developed from field test results by Robertson and Campanella (1983). Anagnostopoulos et al. (2003) provided a similar relationship correlating qc, N60 , and D50. Or ¢ qc ≤ pa N60 5 7.6429D0.26 50 (2.50) where pa 5 atmospheric pressure (same unit as qc). Correlations of Soil Types Robertson and Campanella (1983) provided the correlations shown in Figure 2.31 between qc and the friction ratio [Eq. (2.41)] to identify various types of soil encountered in the field. Correlations for Undrained Shear Strength (cu), Preconsolidation Pressure (scr ), and Overconsolidation Ratio (OCR) for Clays The undrained shear strength, cu, can be expressed as cu 5 qc 2 so NK (2.51) 106 Chapter 2: Natural Soil Deposits and Subsoil Exploration Cone point resistance, qc (MN/m2) 40 Sands 20 Silty sands 10 8 6 4 Sandy Clayey silts silts and and silts silty clays Clays 2 1 0.8 0.6 0.4 Peat 0.2 0.1 0 1 2 3 4 5 Friction ratio, Fr (%) 6 Figure 2.31 Robertson and Campanella’s correlation (1983) between qc, Fr, and the type of soil (Robertson and Campanella, 1983) where so 5 total vertical stress NK 5 bearing capacity factor The bearing capacity factor, NK, may vary from 11 to 19 for normally consolidated clays and may approach 25 for overconsolidated clay. According to Mayne and Kemper (1988) NK 5 15 (for electric cone) and NK 5 20 (for mechanical cone) Based on tests in Greece, Anagnostopoulos et al. (2003) determined NK 5 17.2 (for electric cone) and NK 5 18.9 (for mechanical cone) These field tests also showed that cu 5 fc (for mechanical cones) 1.26 (2.52) and cu 5 fc (for electrical cones) (2.53) Mayne and Kemper (1988) provided correlations for preconsolidation pressure (scr) and overconsolidation ratio (OCR) as scr 5 0.243(qc ) 0.96 c c 2 MN>m MN>m2 (2.54) 2.22 Pressuremeter Test (PMT) 107 and OCR 5 0.37¢ qc 2 so 1.01 ≤ sor (2.55) where so and sor 5 total and effective stress, respectively. 2.22 Pressuremeter Test (PMT) The pressuremeter test is an in situ test conducted in a borehole. It was originally developed by Menard (1956) to measure the strength and deformability of soil. It has also been adopted by ASTM as Test Designation 4719. The Menard-type PMT consists essentially of a probe with three cells. The top and bottom ones are guard cells and the middle one is the measuring cell, as shown schematically in Figure 2.32a. The test is conducted in a prebored hole with a diameter that is between 1.03 and 1.2 times the nominal diameter of the probe. The probe that is most commonly used has a diameter of 58 mm and a length of 420 mm. The probe cells can be expanded by either liquid or gas. The guard cells are expanded to reduce the end-condition effect on the measuring cell, which has a volume (Vo ) of 535 cm3. Following are the dimensions for the probe diameter and the diameter of the borehole, as recommended by ASTM: Probe diameter (mm) Nominal (mm) Maximum (mm) 44 58 74 45 60 76 53 70 89 Gas/water line Borehole diameter pl Pressure, p Zone I Zone II Zone III pf Guard cell p Measuring cell Guard cell po v Vo (a) Vo υo Vo υm Vo υf (b) 2(Vo υo) Figure 2.32 (a) Pressuremeter; (b) plot of pressure versus total cavity volume Total cavity volume, V 108 Chapter 2: Natural Soil Deposits and Subsoil Exploration In order to conduct a test, the measuring cell volume, Vo, is measured and the probe is inserted into the borehole. Pressure is applied in increments and the new volume of the cell is measured. The process is continued until the soil fails or until the pressure limit of the device is reached. The soil is considered to have failed when the total volume of the expanded cavity (V) is about twice the volume of the original cavity. After the completion of the test, the probe is deflated and advanced for testing at another depth. The results of the pressuremeter test are expressed in the graphical form of pressure versus volume, as shown in Figure 2.32b. In the figure, Zone I represents the reloading portion during which the soil around the borehole is pushed back into the initial state (i.e., the state it was in before drilling). The pressure po represents the in situ total horizontal stress. Zone II represents a pseudoelastic zone in which the cell volume versus cell pressure is practically linear. The pressure pf represents the creep, or yield, pressure. The zone marked III is the plastic zone. The pressure pl represents the limit pressure. Figure 2.33 shows some photographs for a pressuremeter test in the field. The pressuremeter modulus, Ep, of the soil is determined with the use of the theory of expansion of an infinitely thick cylinder. Thus, Ep 5 2(1 1 ms ) (Vo 1 vm ) ¢ Dp ≤ Dv (2.56) where vm 5 vo 1 vf 2 Dp 5 pf 2 po Dv 5 vf 2 vo ms 5 Poisson’s ratio (which may be assumed to be 0.33) The limit pressure pl is usually obtained by extrapolation and not by direct measurement. In order to overcome the difficulty of preparing the borehole to the proper size, selfboring pressuremeters (SBPMTs) have also been developed. The details concerning SBPMTs can be found in the work of Baguelin et al. (1978). Correlations between various soil parameters and the results obtained from the pressuremeter tests have been developed by various investigators. Kulhawy and Mayne (1990) proposed that, for clays, scr 5 0.45pl (2.57) where scr 5 preconsolidation pressure. On the basis of the cavity expansion theory, Baguelin et al. (1978) proposed that cu 5 (pl 2 po ) Np (2.58) 2.22 Pressuremeter Test (PMT) (a) (c) (b) (d) 109 Figure 2.33 Pressuremeter test in the field: (a) the pressuremeter probe; (b) drilling the bore hole by wet rotary method; (c) pressuremeter control unit with probe in the background; (d) getting ready to insert the pressuremeter probe into the bore hole (Courtesy of Jean-Louis Briaud, Texas A&M University, College Station, Texas) 110 Chapter 2: Natural Soil Deposits and Subsoil Exploration where cu 5 undrained shear strength of a clay Ep Np 5 1 1 ln ¢ ≤ 3cu Typical values of Np vary between 5 and 12, with an average of about 8.5. Ohya et al. (1982) (see also Kulhawy and Mayne, 1990) correlated Ep with field standard penetration numbers (N60 ) for sand and clay as follows: Clay: Ep (kN>m2 ) 5 1930N 0.63 60 2 Sand: Ep (kN>m ) 5 2.23 908N 0.66 60 (2.59) (2.60) Dilatometer Test The use of the flat-plate dilatometer test (DMT) is relatively recent (Marchetti, 1980; Schmertmann, 1986). The equipment essentially consists of a flat plate measuring 220 mm (length) 3 95 mm (width) 3 14 mm (thickness). A thin, flat, circular, expandable steel membrane having a diameter of 60 mm is located flush at the center on one side of the plate (Figure 2.34a). Figure 2.35 shows two flat-plate dilatometers with other instruments for conducting a test in the field. The dilatometer probe is inserted into the ground with a cone penetrometer testing rig (Figure 2.34b). Gas and electric lines extend from the surface control box, through the penetrometer rod, and into the blade. At the required depth, high-pressure nitrogen gas is used to inflate the membrane. Two pressure readings are taken: 1. The pressure A required to “lift off” the membrane. 2. The pressure B at which the membrane expands 1.1 mm into the surrounding soil 60 mm 95 mm (a) (b) Figure 2.34 (a) Schematic diagram of a flat-plate dilatometer; (b) dilatometer probe inserted into ground 2.23 Dilatometer Test 111 Figure 2.35 Dilatometer and other equipment (Courtesy of N. Sivakugan, James Cook University, Australia) The A and B readings are corrected as follows (Schmertmann, 1986): Contact stress, po 5 1.05(A 1 DA 2 Zm ) 2 0.05(B 2 DB 2 Zm ) Expansion stress, p1 5 B 2 Zm 2 DB (2.61) (2.62) where DA 5 vacuum pressure required to keep the membrane in contact with its seating DB 5 air pressure required inside the membrane to deflect it outward to a center expansion of 1.1 mm Zm 5 gauge pressure deviation from zero when vented to atmospheric pressure The test is normally conducted at depths 200 to 300 mm apart. The result of a given test is used to determine three parameters: 1. Material index, ID 5 p1 2 po po 2 uo 2. Horizontal stress index, KD 5 po 2 uo sor 3. Dilatometer modulus, ED (kN>m2 ) 5 34.7(p1 kN>m2 2 po kN>m2 ) 112 Chapter 2: Natural Soil Deposits and Subsoil Exploration where uo 5 pore water pressure sor 5 in situ vertical effective stress Figure 2.36 shows the results of a dilatometer test conducted in Bangkok soft clay and reported by Shibuya and Hanh (2001). Based on his initial tests, Marchetti (1980) provided the following correlations. KD 0.47 ≤ 2 0.6 1.5 (2.63) OCR 5 (0.5KD ) 1.56 (2.64) Ko 5 ¢ cu 5 0.22 sor 0 pO , p1 (kN/m2) 300 600 0 ID 0.3 (for normally consolidated clay) 0.6 0 KD 3 6 0 (2.65) ED (kN/m2) 2,000 4,000 5,000 0 2 4 Depth (m) 6 8 p1 10 pO 12 14 Figure 2.36 A dilatometer test result conducted on soft Bangkok clay (Redrawn from Shibuya and Hanh, 2001) 2.24 Coring of Rocks ¢ 113 cu cu ≤ 5 ¢ ≤ (0.5KD ) 1.25 sor OC sor NC (2.66) Es 5 (1 2 m2s )ED (2.67) where Ko 5 coefficient of at-rest earth pressure OCR 5 overconsolidation ratio OC 5 overconsolidated soil NC 5 normally consolidated soil Es 5 modulus of elasticity Other relevant correlations using the results of dilatometer tests are as follows: • For undrained cohesion in clay (Kamei and Iwasaki, 1995): cu 5 0.35 s0r (0.47KD ) 1.14 • (2.68) For soil friction angle (ML and SP-SM soils) (Ricceri et al., 2002): fr 5 31 1 KD 0.236 1 0.066KD fult r 5 28 1 14.6 logKD 2 2.1(logKD ) 2 (2.69a) (2.69b) Schmertmann (1986) also provided a correlation between the material index (ID ) and the dilatometer modulus (ED ) for a determination of the nature of the soil and its unit weight (g). This relationship is shown in Figure 2.37. 2.24 Coring of Rocks When a rock layer is encountered during a drilling operation, rock coring may be necessary. To core rocks, a core barrel is attached to a drilling rod. A coring bit is attached to the bottom of the barrel (Fig. 2.38). The cutting elements may be diamond, tungsten, carbide, and so on. Table 2.10 summarizes the various types of core barrel and their sizes, as well as the compatible drill rods commonly used for exploring foundations. The coring is advanced by rotary drilling. Water is circulated through the drilling rod during coring, and the cutting is washed out. 114 Chapter 2: Natural Soil Deposits and Subsoil Exploration 200 Clay Silt Silty Clayey Sand Sandy Silty 100 Dilatometer modulus, ED (MN/m2) gid Ri 0) 0 (2. e ns de ry 0) e V 2.1 ( 50 5 ty idi rig w Lo 1.80) ( ty nsi de m diu .80) e M (1 cy ten sis on ) c gh 1.90 ( Hi cy ten sis n o c ) m diu (1.80 Me cy ten sis n o wc 0)* Lo (1.7 10 ity id rig um 0) i d Me (1.9 nse De 5) 9 (1. rd Ha 5) 0 (2. 20 id rig ry Ve .15) (2 ose Lo 0) 7 (1. ity ens w d 0) o L 7 (1. le sib res p ) m Co (1.60 ft So )* 60 (1. 2 1.2 1.0 0.35 0.6 Mud/Peat (1.50) 0.5 10 0.2 0.5 0.9 3.3 1.2 1.8 (␥) – Approximate soil unit weight in t/m3 shown in parentheses * – If PI > 50, then ␥ in these regions is overestimated by about 0.10 t/m3 1 2 Material index, ID 5 10 Figure 2.37 Chart for determination of soil description and unit weight (After Schmertmann, 1986) (Note: 1 t>m3 5 9.81 kN>m3) (Schmertmann, J. H. (1986). “Suggested method for performing the flat dilatometer test,” Geotechnical Testing Journal, ASTM, Vol. 9, No. 2, pp. 93-101, Fig. 2. Copyright ASTM INTERNATIONAL. Reprinted with permission.) Table 2.10 Standard Size and Designation of Casing, Core Barrel, and Compatible Drill Rod Casing and core barrel designation Outside diameter of core barrel bit (mm) EX AX BX NX 36.51 47.63 58.74 74.61 Drill rod designation Outside diameter of drill rod (mm) Diameter of borehole (mm) Diameter of core sample (mm) E A B N 33.34 41.28 47.63 60.33 38.1 50.8 63.5 76.2 22.23 28.58 41.28 53.98 Two types of core barrel are available: the single-tube core barrel (Figure 2.38a) and the double-tube core barrel (Figure 2.38b). Rock cores obtained by single-tube core barrels can be highly disturbed and fractured because of torsion. Rock cores smaller than the BX size tend to fracture during the coring process. Figure 2.39 shows 2.24 Coring of Rocks Drill rod Drill rod Inner barrel Core barrel Rock Rock Rock core 115 Outer barrel Rock Rock core Core lifter Coring bit Core lifter Coring bit (a) (b) Figure 2.38 Rock coring: (a) singletube core barrel; (b) double-tube core barrel Figure 2.39 Diamond coring bit (Courtesy of Braja M. Das, Henderson, NV) 116 Chapter 2: Natural Soil Deposits and Subsoil Exploration (a) (b) Figure 2.40 Diamond coring bit attached to a double-tube core barrel: (a) end view; (b) side view (Courtesy of Professional Service Industries, Inc. (PSI), Waukesha, Wisconsin) the photograph of a diamond coring bit. Figure 2.40 shows the end and side views of a diamond coring bit attached to a double-tube core barrel. When the core samples are recovered, the depth of recovery should be properly recorded for further evaluation in the laboratory. Based on the length of the rock core 2.25 Preparation of Boring Logs 117 recovered from each run, the following quantities may be calculated for a general evaluation of the rock quality encountered: Recovery ratio 5 length of core recovered theoretical length of rock cored (2.70) Rock quality designation (RQD) 5 S length of recovered pieces equal to or larger than 101.6 mm theoretical length of rock cored (2.71) A recovery ratio of unity indicates the presence of intact rock; for highly fractured rocks, the recovery ratio may be 0.5 or smaller. Table 2.11 presents the general relationship (Deere, 1963) between the RQD and the in situ rock quality. Table 2.11 Relation between in situ Rock Quality and RQD 2.25 RQD Rock quality 0– 0.25 0.25– 0.5 0.5–0.75 0.75– 0.9 0.9–1 Very poor Poor Fair Good Excellent Preparation of Boring Logs The detailed information gathered from each borehole is presented in a graphical form called the boring log. As a borehole is advanced downward, the driller generally should record the following information in a standard log: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Name and address of the drilling company Driller’s name Job description and number Number, type, and location of boring Date of boring Subsurface stratification, which can be obtained by visual observation of the soil brought out by auger, split-spoon sampler, and thin-walled Shelby tube sampler Elevation of water table and date observed, use of casing and mud losses, and so on Standard penetration resistance and the depth of SPT Number, type, and depth of soil sample collected In case of rock coring, type of core barrel used and, for each run, the actual length of coring, length of core recovery, and RQD This information should never be left to memory, because doing so often results in erroneous boring logs. 118 Chapter 2: Natural Soil Deposits and Subsoil Exploration Boring Log Name of the Project Two-story apartment building Location Johnson & Olive St. Date of Boring March 2, 2005 60.8 m Boring No. 3 Type of Hollow-stem auger Ground Boring Elevation Soil wn Soil Depth sample N Comments 60 (%) description (m) type and number Light brown clay (fill) 1 Silty sand (SM) G.W.T. 3.5 m Light gray clayey silt (ML) 2 3 9 8.2 SS-2 12 17.6 LL 38 PI 11 20.4 LL 36 qu 112 kN/m2 4 5 6 Sand with some gravel (SP) End of boring @ 8 m SS-1 ST-1 SS-3 11 20.6 7 SS-4 27 8 N60 standard penetration number wn natural moisture content LL liquid limit; PI plasticity index qu unconfined compression strength SS split-spoon sample; ST Shelby tube sample 9 Groundwater table observed after one week of drilling Figure 2.41 A typical boring log After completion of the necessary laboratory tests, the geotechnical engineer prepares a finished log that includes notes from the driller’s field log and the results of tests conducted in the laboratory. Figure 2.41 shows a typical boring log. These logs have to be attached to the final soil-exploration report submitted to the client. The figure also lists the classifications of the soils in the left-hand column, along with the description of each soil (based on the Unified Soil Classification System). 2.26 Geophysical Exploration Several types of geophysical exploration techniques permit a rapid evaluation of subsoil characteristics. These methods also allow rapid coverage of large areas and are less expensive than conventional exploration by drilling. However, in many cases, definitive interpretation of the results is difficult. For that reason, such techniques should be used for preliminary work only. Here, we discuss three types of geophysical exploration technique: the seismic refraction survey, cross-hole seismic survey, and resistivity survey. Seismic Refraction Survey Seismic refraction surveys are useful in obtaining preliminary information about the thickness of the layering of various soils and the depth to rock or hard soil at a site. 2.26 Geophysical Exploration 119 Refraction surveys are conducted by impacting the surface, such as at point A in Figure 2.42a, and observing the first arrival of the disturbance (stress waves) at several other points (e.g., B, C, D, c ). The impact can be created by a hammer blow or by a small explosive charge. The first arrival of disturbance waves at various points can be recorded by geophones. The impact on the ground surface creates two types of stress wave: P waves (or plane waves) and S waves (or shear waves). P waves travel faster than S waves; hence, the first arrival of disturbance waves will be related to the velocities of the P waves in various layers. The velocity of P waves in a medium is v5 Es g ã ¢g≤ (1 2 ms ) (1 2 2ms ) (1 1 ms ) where Es 5 modulus of elasticity of the medium g 5 unit weight of the medium g 5 acceleration due to gravity ms 5 Poisson’s ratio x (x1) B A v1 Layer I v1 (x2) C v1 (x3) D v1 v1 Z1 Velocity v1 v2 v2 Layer II v2 v3 Layer III Velocity v3 Time of first arrival (a) d c Ti2 b Ti1 xc a Distance, x (b) Figure 2.42 Seismic refraction survey Z2 Velocity v2 (2.72) 120 Chapter 2: Natural Soil Deposits and Subsoil Exploration To determine the velocity v of P waves in various layers and the thicknesses of those layers, we use the following procedure: Step 1. Obtain the times of first arrival, t1, t2, t3, c, at various distances x1, x2, x3, c from the point of impact. Step 2. Plot a graph of time t against distance x. The graph will look like the one shown in Figure 2.42b. Step 3. Determine the slopes of the lines ab, bc, cd, c: Slope of ab 5 1 v1 Slope of bc 5 1 v2 Slope of cd 5 1 v3 Here, v1, v2, v3, c are the P-wave velocities in layers I, II, III, c, respectively (Figure 2.42a). Step 4. Determine the thickness of the top layer: Z1 5 1 v2 2 v1 x 2 Å v2 1 v1 c (2.73) The value of xc can be obtained from the plot, as shown in Figure 2.42b. Step 5. Determine the thickness of the second layer: "v23 2 v21 v3v2 1 Z2 5 BTi2 2 2Z1 R v3v1 2 "v23 2 v22 (2.74) Here, Ti2 is the time intercept of the line cd in Figure 2.42b, extended backwards. (For detailed derivatives of these equations and other related information, see Dobrin, 1960, and Das, 1992). The velocities of P waves in various layers indicate the types of soil or rock that are present below the ground surface. The range of the P-wave velocity that is generally encountered in different types of soil and rock at shallow depths is given in Table 2.12. In analyzing the results of a refraction survey, two limitations need to be kept in mind: 1. The basic equations for the survey—that is, Eqs. (2.73) and (2.74)—are based on the assumption that the P-wave velocity v1 , v2 , v3 , c. 2. When a soil is saturated below the water table, the P-wave velocity may be deceptive. P waves can travel with a velocity of about 1500 m>sec through water. For dry, loose soils, the velocity may be well below 1500 m>sec. However, in a saturated condition, the waves will travel through water that is present in the void spaces with a velocity of about 1500 m>sec. If the presence of groundwater has not been detected, the P-wave velocity may be erroneously interpreted to indicate a stronger material (e.g., sandstone) than is actually present in situ. In general, geophysical interpretations should always be verified by the results obtained from borings. 2.26 Geophysical Exploration 121 Table 2.12 Range of P-Wave Velocity in Various Soils and Rocks P-wave velocity m , sec Type of soil or rock Soil Sand, dry silt, and fine-grained topsoil Alluvium Compacted clays, clayey gravel, and dense clayey sand Loess 200–1000 500–2000 1000–2500 Slate and shale Sandstone Granite Sound limestone 2500–5000 1500–5000 4000–6000 5000–10,000 250–750 Rock Example 2.1 The results of a refraction survey at a site are given in the following table: Distance of geophone from the source of disturbance (m) Time of first arrival (sec 3 103) 2.5 5 7.5 10 15 20 25 30 35 40 50 11.2 23.3 33.5 42.4 50.9 57.2 64.4 68.6 71.1 72.1 75.5 Determine the P-wave velocities and the thickness of the material encountered. Solution Velocity In Figure 2.43, the times of first arrival of the P waves are plotted against the distance of the geophone from the source of disturbance. The plot has three straight-line segments. The velocity of the top three layers can now be calculated as follows: Slope of segment 0a 5 1 time 23 3 1023 5 5 v1 distance 5.25 122 Chapter 2: Natural Soil Deposits and Subsoil Exploration 80 Time of first arrival, t (103)—in seconds c b Ti2 = 65 10–3 sec 3.5 14.75 60 13.5 a 11 23 xc = 10.5 m 40 20 5.25 0 0 10 20 30 Distance, x (m) 40 50 Figure 2.43 Plot of first arrival time of P wave versus distance of geophone from source of disturbance or v1 5 5.25 3 103 5 228 m , sec (top layer) 23 Slope of segment ab 5 1 13.5 3 10 23 5 v2 11 or v2 5 11 3 103 5 814.8 m , sec (middle layer) 13.5 Slope of segment bc 5 1 3.5 3 10 23 5 v3 14.75 or v3 5 4214 m , sec (third layer) Comparing the velocities obtained here with those given in Table 2.12 indicates that the third layer is a rock layer. Thickness of Layers From Figure 2.43, xc 5 10.5 m, so Z1 5 1 v2 2 v1 x 2Å v2 1 v1 c Thus, Z1 5 1 814.8 2 228 3 10.5 5 3.94 m 2Å 814.8 1 228 123 2.26 Geophysical Exploration Again, from Eq. (2.74) 2Z1"v23 2 v21 (v3 ) (v2 ) 1 Z2 5 BTi2 2 R 2 (v3v1 ) "v23 2 v22 The value of Ti2 (from Figure 2.43) is 65 3 10 23 sec. Hence, 2(3.94)"(4214) 2 2 (228) 2 (4214) (814.8) 1 Z2 5 B65 3 10 23 2 R 2 (4214) (228) "(4214) 2 2 (814.8) 2 1 5 (0.065 2 0.0345)830.47 5 12.66 m 2 Thus, the rock layer lies at a depth of Z1 1 Z2 5 3.94 1 12.66 5 16.60 m from the surface of the ground. ■ Cross-Hole Seismic Survey The velocity of shear waves created as the result of an impact to a given layer of soil can be effectively determined by the cross-hole seismic survey (Stokoe and Woods, 1972). The principle of this technique is illustrated in Figure 2.44, which shows two holes drilled into the ground a distance L apart. A vertical impulse is created at the bottom of one borehole by means of an impulse rod. The shear waves thus generated are recorded by a vertically sensitive transducer. The velocity of shear waves can be calculated as vs 5 L t (2.75) where t 5 travel time of the waves. Impulse Oscilloscope Vertical velocity transducer Vertical velocity transducer Shear wave L Figure 2.44 Cross-hole method of seismic survey 124 Chapter 2: Natural Soil Deposits and Subsoil Exploration The shear modulus Gs of the soil at the depth at which the test is taken can be determined from the relation vs 5 Gs Å (g>g) or Gs 5 v2s g g (2.76) where vs 5 velocity of shear waves g 5 unit weight of soil g 5 acceleration due to gravity The shear modulus is useful in the design of foundations to support vibrating machinery and the like. Resistivity Survey Another geophysical method for subsoil exploration is the electrical resistivity survey. The electrical resistivity of any conducting material having a length L and an area of cross section A can be defined as r5 RA L (2.77) where R 5 electrical resistance. The unit of resistivity is ohm-centimeter or ohm-meter. The resistivity of various soils depends primarily on their moisture content and also on the concentration of dissolved ions in them. Saturated clays have a very low resistivity; dry soils and rocks have a high resistivity. The range of resistivity generally encountered in various soils and rocks is given in Table 2.13. The most common procedure for measuring the electrical resistivity of a soil profile makes use of four electrodes driven into the ground and spaced equally along a straight line. The procedure is generally referred to as the Wenner method (Figure 2.45a). Table 2.13 Representative Values of Resistivity Material Resistivity (ohm ? m) Sand Clays, saturated silt Clayey sand Gravel Weathered rock Sound rock 500–1500 0–100 200–500 1500–4000 1500–2500 .5000 2.26 Geophysical Exploration 125 I V d d d Layer 1 Resistivity, 1 Z1 Layer 2 Resistivity, 2 (a) Slope 2 Slope 1 Z1 d (b) Figure 2.45 Electrical resistivity survey: (a) Wenner method; (b) empirical method for determining resistivity and thickness of each layer The two outside electrodes are used to send an electrical current I (usually a dc current with nonpolarizing potential electrodes) into the ground. The current is typically in the range of 50 to 100 milliamperes. The voltage drop, V, is measured between the two inside electrodes. If the soil profile is homogeneous, its electrical resistivity is r5 2pdV I (2.78) In most cases, the soil profile may consist of various layers with different resistivities, and Eq. (2.78) will yield the apparent resistivity. To obtain the actual resistivity of various layers and their thicknesses, one may use an empirical method that involves conducting tests at various electrode spacings (i.e., d is changed). The sum of the apparent resistivities, Sr, is plotted against the spacing d, as shown in Figure 2.45b. The plot thus obtained has relatively straight segments, the slopes of which give the resistivity of individual layers. The thicknesses of various layers can be estimated as shown in Figure 2.45b. The resistivity survey is particularly useful in locating gravel deposits within a finegrained soil. 126 Chapter 2: Natural Soil Deposits and Subsoil Exploration 2.27 Subsoil Exploration Report At the end of all soil exploration programs, the soil and rock specimens collected in the field are subject to visual observation and appropriate laboratory testing. (The basic soil tests were described in Chapter 1.) After all the required information has been compiled, a soil exploration report is prepared for use by the design office and for reference during future construction work. Although the details and sequence of information in such reports may vary to some degree, depending on the structure under consideration and the person compiling the report, each report should include the following items: 1. A description of the scope of the investigation 2. A description of the proposed structure for which the subsoil exploration has been conducted 3. A description of the location of the site, including any structures nearby, drainage conditions, the nature of vegetation on the site and surrounding it, and any other features unique to the site 4. A description of the geological setting of the site 5. Details of the field exploration—that is, number of borings, depths of borings, types of borings involved, and so on 6. A general description of the subsoil conditions, as determined from soil specimens and from related laboratory tests, standard penetration resistance and cone penetration resistance, and so on 7. A description of the water-table conditions 8. Recommendations regarding the foundation, including the type of foundation recommended, the allowable bearing pressure, and any special construction procedure that may be needed; alternative foundation design procedures should also be discussed in this portion of the report 9. Conclusions and limitations of the investigations The following graphical presentations should be attached to the report: 1. A site location map 2. A plan view of the location of the borings with respect to the proposed structures and those nearby 3. Boring logs 4. Laboratory test results 5. Other special graphical presentations The exploration reports should be well planned and documented, as they will help in answering questions and solving foundation problems that may arise later during design and construction. Problems 2.1 2.2 For a Shelby tube, given: outside diameter 5 76.2 mm and inside diameter 73 mm in. What is the area ratio of the tube? A soil profile is shown in Figure P2.2 along with the standard penetration numbers in the clay layer. Use Eqs. (2.8) and (2.9) to determine the variation of cu and OCR with depth. What is the average value of cu and OCR? Problems 1.5 m Groundwater table 1.5 m N60 5 1.5 m 8 1.5 m A 8 127 Dry sand ␥ 16.5 kN/m3 Sand ␥sat 19 kN/m3 Clay ␥sat 16.8 kN/m3 1.5 m 9 1.5 m 10 Sand Figure P2.2 2.3 2.4 2.5 2.6 2.7 Following is the variation of the field standard penetration number (N60 ) in a sand deposit: Depth (m) N60 1.5 3 4.5 6 7.9 9 6 8 9 8 13 14 The groundwater table is located at a depth of 6 m. Given: the dry unit weight of sand from 0 to a depth of 6 m is 18 kN>m3, and the saturated unit weight of sand for depth 6 to 12 m is 20.2 kN>m3. Use the relationship of Skempton given in Eq. (2.12) to calculate the corrected penetration numbers. For the soil profile described in Problem 2.3, estimate an average peak soil friction angle. Use Eq. (2.28). Repeat Problem 2.4 using Eq. (2.27). Refer to Problem 2.3. Using Eq. (2.20), determine the average relative density of sand. The following table gives the variation of the field standard penetration number (N60 ) in a sand deposit: Depth (m) N60 1.5 3.0 4.5 6.0 7.5 9.0 5 11 14 18 16 21 128 Chapter 2: Natural Soil Deposits and Subsoil Exploration 2.8 The groundwater table is located at a depth of 12 m. The dry unit weight of sand from 0 to a depth of 12 m is 17.6 kN>m3. Assume that the mean grain size (D50 ) of the sand deposit to be about 0.8 mm. Estimate the variation of the relative density with depth for sand. Use Eq. (2.21). Following are the standard penetration numbers determined from a sandy soil in the field: Depth (m) Unit weight of soil (kN , m 3) N60 3.0 4.5 6.0 7.5 9.0 10.5 12.0 16.66 16.66 16.66 18.55 18.55 18.55 18.55 7 9 11 16 18 20 22 Using Eq. (2.27), determine the variation of the peak soil friction angle, fr. Estimate an average value of fr for the design of a shallow foundation. (Note: For depth greater than 6 m, the unit weight of soil is 18.55 kN>m3.) 2.9 Refer to Problem 2.8. Assume that the sand is clean and normally consolidated. Estimate the average value of the modulus of elasticity between depths of 6 m and 9 m. 2.10 Following are the details for a soil deposit in sand: Depth (m) 3.0 4.5 6.0 2.11 2.12 2.13 2.14 Effective overburden pressure (kN , m 2) 55 82 98 Field standard penetration number, N60 9 11 12 Assume the uniformity coefficient (Cu ) of the sand to be 2.8 and an overconsolidation ratio (OCR) of 2. Estimate the average relative density of the sand between the depth of 3 to 6 m. Use Eq. (2.19). Refer to Figure P2.2. Vane shear tests were conducted in the clay layer. The vane dimensions were 63.5 mm (D) 3 127 mm (H). For the test at A, the torque required to cause failure was 0.051 N # m. For the clay, given: liquid limit 5 46 and plastic limit 5 21. Estimate the undrained cohesion of the clay for use in the design by using Bjerrum’s l relationship [Eq. (2.35a)]. Refer to Problem 2.11. Estimate the overconsolidation ratio of the clay. Use Eqs. (2.37) and (2.38). a. A vane shear test was conducted in a saturated clay. The height and diameter of the vane were 101.6 mm and 50.8 mm, respectively. During the test, the maximum torque applied was 23 lb-ft. Determine the undrained shear strength of the clay. b. The clay soil described in part (a) has a liquid limit of 58 and a plastic limit of 29. What would be the corrected undrained shear strength of the clay for design purposes? Use Bjerrum’s relationship for l [Eq. (2.35a)]. Refer to Problem 2.13. Determine the overconsolidation ratio for the clay. Use Eqs. (2.37) and (2.40). Use s0r 5 64.2 kN>m2. Problems 129 2.15 In a deposit of normally consolidated dry sand, a cone penetration test was conducted. Following are the results: 2.16 2.17 2.18 2.19 Depth (m) Point resistance of cone, qc (MN , m2) 1.5 3.0 4.5 6.0 7.5 9.0 2.06 4.23 6.01 8.18 9.97 12.42 Assuming the dry unit weight of sand to be 16 kN>m3, estimate the average peak friction angle, fr, of the sand. Use Eq. (2.48). Refer to Problem 2.15. Using Eq. (2.46), determine the variation of the relative density with depth. In the soil profile shown in Figure P2.17, if the cone penetration resistance (qc) at A (as determined by an electric friction-cone penetrometer) is 0.8 MN> m.2, estimate a. The undrained cohesion, cu b. The overconsolidation ratio, OCR In a pressuremeter test in a soft saturated clay, the measuring cell volume Vo 5 535 cm3, po 5 42.4 kN>m2, pf 5 326.5 kN>m2, vo 5 46 cm3, and vf 5 180 cm3. Assuming Poisson’s ratio (ms ) to be 0.5 and using Figure 2.32, calculate the pressuremeter modulus (Ep ). A dilatometer test was conducted in a clay deposit. The groundwater table was located at a depth of 3 m below the surface. At a depth of 8 m below the surface, the contact pressure (po ) was 280 kN>m2 and the expansion stress (p1 ) was 350 kN>m2. Determine the following: a. Coefficient of at-rest earth pressure, Ko b. Overconsolidation ratio, OCR c. Modulus of elasticity, Es Assume sor at a depth of 8 m to be 95 kN>m2 and ms 5 0.35. 2m Water table Clay ␥ 18 kN/m3 Clay ␥sat 20 kN/m3 4m A Figure P2.17 130 Chapter 2: Natural Soil Deposits and Subsoil Exploration 2.20 A dilatometer test was conducted in a sand deposit at a depth of 6 m. The groundwater table was located at a depth of 2 m below the ground surface. Given, for the sand: gd 5 14.5 kN>m3 and gsat 5 19.8 kN>m3. The contact stress during the test was 260 kN>m2. Estimate the soil friction angle, fr. 2.21 The P-wave velocity in a soil is 1900 m>sec. Assuming Poisson’s ratio to be 0.32, calculate the modulus of elasticity of the soil. Assume that the unit weight of soil is 18 kN>m3. 2.22 The results of a refraction survey (Figure 2.42a) at a site are given in the following table. Determine the thickness and the P-wave velocity of the materials encountered. Distance from the source of disturbance (m) Time of first arrival of P- waves (sec 3 103) 2.5 5.0 7.5 10.0 15.0 20.0 25.0 30.0 40.0 50.0 5.08 10.16 15.24 17.01 20.02 24.2 27.1 28.0 31.1 33.9 References AMERICAN SOCIETY FOR TESTING AND MATERIALS (2001). Annual Book of ASTM Standards, Vol. 04.08, West Conshohocken, PA. AMERICAN SOCIETY OF CIVIL ENGINEERS (1972). “Subsurface Investigation for Design and Construction of Foundations of Buildings,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 98, No. SM5, pp. 481–490. ANAGNOSTOPOULOS, A., KOUKIS, G., SABATAKAKIS, N., and TSIAMBAOS, G. (2003). “Empirical Correlations of Soil Parameters Based on Cone Penetration Tests (CPT) for Greek Soils,” Geotechnical and Geological Engineering, Vol. 21, No. 4, pp. 377–387. BAGUELIN, F., JÉZÉQUEL, J. F., and SHIELDS, D. H. (1978). The Pressuremeter and Foundation Engineering, Trans Tech Publications, Clausthal, Germany. BALDI, G., BELLOTTI, R., GHIONNA, V., and JAMIOLKOWSKI, M. (1982). “Design Parameters for Sands from CPT,” Proceedings, Second European Symposium on Penetration Testing, Amsterdam, Vol. 2, pp. 425–438. BAZARAA, A. (1967). 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SHIBUYA, S., and HANH, L. T. (2001). “Estimating Undrained Shear Strength of Soft Clay Ground Improved by Pre-Loading with PVD—Case History in Bangkok,” Soils and Foundations, Vol. 41, No. 4, pp. 95–101. SKEMPTON, A. W. (1986). “Standard Penetration Test Procedures and the Effect in Sands of Overburden Pressure, Relative Density, Particle Size, Aging and Overconsolidation,” Geotechnique, Vol. 36, No. 3, pp. 425–447. SOWERS, G. B., and SOWERS, G. F. (1970). Introductory Soil Mechanics and Foundations, 3d ed., Macmillan, New York. STOKOE, K. H., and WOODS, R. D. (1972). “In Situ Shear Wave Velocity by Cross-Hole Method,” Journal of Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 98, No. SM5, pp. 443– 460. SZECHY, K., and VARGA, L. (1978). Foundation Engineering—Soil Exploration and Spread Foundation, Akademiai Kiado, Budapest, Hungary. WOLFF, T. F. (1989). “Pile Capacity Prediction Using Parameter Functions,” in Predicted and Observed Axial Behavior of Piles, Results of a Pile Prediction Symposium, sponsored by the Geotechnical Engineering Division, ASCE, Evanston, IL, June, 1989, ASCE Geotechnical Special Publication No. 23, pp. 96–106. 3 3.1 Shallow Foundations: Ultimate Bearing Capacity Introduction To perform satisfactorily, shallow foundations must have two main characteristics: 1. They have to be safe against overall shear failure in the soil that supports them. 2. They cannot undergo excessive displacement, or settlement. (The term excessive is relative, because the degree of settlement allowed for a structure depends on several considerations.) The load per unit area of the foundation at which shear failure in soil occurs is called the ultimate bearing capacity, which is the subject of this chapter. 3.2 General Concept Consider a strip foundation with a width of B resting on the surface of a dense sand or stiff cohesive soil, as shown in Figure 3.1a. Now, if a load is gradually applied to the foundation, settlement will increase. The variation of the load per unit area on the foundation (q) with the foundation settlement is also shown in Figure 3.1a. At a certain point—when the load per unit area equals qu—a sudden failure in the soil supporting the foundation will take place, and the failure surface in the soil will extend to the ground surface. This load per unit area, qu, is usually referred to as the ultimate bearing capacity of the foundation. When such sudden failure in soil takes place, it is called general shear failure. If the foundation under consideration rests on sand or clayey soil of medium compaction (Figure 3.1b), an increase in the load on the foundation will also be accompanied by an increase in settlement. However, in this case the failure surface in the soil will gradually extend outward from the foundation, as shown by the solid lines in Figure 3.1b. When the load per unit area on the foundation equals qu(1), movement of the foundation will be accompanied by sudden jerks. A considerable movement of the foundation is then required for the failure surface in soil to extend to the ground surface (as shown by the broken lines in the figure). The load per unit area at which this happens is the ultimate bearing capacity, qu. Beyond that point, an increase in load will be 133 134 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity Load/unit area, q B qu Failure surface in soil (a) Settlement Load/unit area, q B qu(1) qu Failure surface (b) Settlement Load/unit area, q B qu(1) qu Failure surface (c) qu Surface footing Settlement Figure 3.1 Nature of bearing capacity failure in soil: (a) general shear failure: (b) local shear failure; (c) punching shear failure (Redrawn after Vesic, 1973) (Vesic, A. S. (1973). “Analysis of Ultimate Loads of Shallow Foundations,” Journal of Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 99, No. SM1, pp. 45–73. With permission from ASCE.) accompanied by a large increase in foundation settlement. The load per unit area of the foundation, qu(1), is referred to as the first failure load (Vesic, 1963). Note that a peak value of q is not realized in this type of failure, which is called the local shear failure in soil. If the foundation is supported by a fairly loose soil, the load–settlement plot will be like the one in Figure 3.1c. In this case, the failure surface in soil will not extend to the ground surface. Beyond the ultimate failure load, qu, the load–settlement plot will be steep and practically linear. This type of failure in soil is called the punching shear failure. Vesic (1963) conducted several laboratory load-bearing tests on circular and rectangular plates supported by a sand at various relative densities of compaction, Dr. The variations of qu(1)> 12gB and qu> 12gB obtained from those tests, where B is the diameter of a circular plate or width of a rectangular plate and g is a dry unit weight of sand, are shown in Figure 3.2. It is important to note from this figure that, for Dr > about 70%, the general shear type of failure in soil occurs. On the basis of experimental results, Vesic (1973) proposed a relationship for the mode of bearing capacity failure of foundations resting on sands. Figure 3.3 3.2 General Concept 0.2 0.3 0.4 Punching shear 700 600 500 Relative density, Dr 0.5 0.6 0.7 0.8 135 0.9 General shear Local shear 400 300 qu qu(1) 100 90 80 70 60 50 qu 1 B 2 and 1 B 2 200 1 B 2 40 Legend Circular plate 203 mm (8 in.) Circular plate 152 mm (6 in.) Circular plate 102 mm (4 in.) Circular plate 51 mm (2 in.) Rectangular plate 51 305 mm (2 12 in.) 30 qu(1) 1 B 2 20 Reduced by 0.6 Small signs indicate first failure load 10 1.32 1.35 1.40 1.45 1.50 Dry unit weight, d Unit weight of water, w 1.55 1.60 Figure 3.2 Variation of qu(1)>0.5gB and qu>0.5gB for circular and rectangular plates on the surface of a sand (Adapted from Vesic, 1963) (From Vesic, A. B. Bearing Capacity of Deep Foundations in Sand. In Highway Research Record 39, Highway Research Board, National Research Council, Washington, D.C., 1963, Figure 28, p. 137. Reproduced with permission of the Transportation Research Board.) shows this relationship, which involves the notation Dr 5 relative density of sand Df 5 depth of foundation measured from the ground surface 2BL B* 5 B1L where B 5 width of foundation L 5 length of foundation (Note: L is always greater than B.) (3.1) 136 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity 0.2 0 Relative density, Dr 0.4 0.6 0.8 1.0 0 1 Punching shear failure Local shear failure General shear failure Df /B* 2 3 4 Df B 5 Figure 3.3 Modes of foundation failure in sand (After Vesic, 1973) (Vesic, A. S. (1973). “Analysis of Ultimate Loads of Shallow Foundations,” Journal of Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 99, No. SM1, pp. 45–73. With permission from ASCE.) For square foundations, B 5 L; for circular foundations, B 5 L 5 diameter, so B* 5 B (3.2) Figure 3.4 shows the settlement S of the circular and rectangular plates on the surface of a sand at ultimate load, as described in Figure 3.2. The figure indicates a general range of S>B with the relative density of compaction of sand. So, in general, we can say that, for foundations at a shallow depth (i.e., small Df> B*), the ultimate load may occur at a foundation settlement of 4 to 10% of B. This condition arises together with general shear failure in soil; however, in the case of local or punching shear failure, the ultimate load may occur at settlements of 15 to 25% of the width of the foundation (B). 3.3 Terzaghi’s Bearing Capacity Theory Terzaghi (1943) was the first to present a comprehensive theory for the evaluation of the ultimate bearing capacity of rough shallow foundations. According to this theory, a foundation is shallow if its depth, Df (Figure 3.5), is less than or equal to its width. Later investigators, however, have suggested that foundations with Df equal to 3 to 4 times their width may be defined as shallow foundations. Terzaghi suggested that for a continuous, or strip, foundation (i.e., one whose widthto-length ratio approaches zero), the failure surface in soil at ultimate load may be assumed to be similar to that shown in Figure 3.5. (Note that this is the case of general shear failure, as defined in Figure 3.1a.) The effect of soil above the bottom of the foundation may also be assumed to be replaced by an equivalent surcharge, q 5 gDf (where g is a unit weight of soil). The failure zone under the foundation can be separated into three parts (see Figure 3.5): 3.3 Terzaghi’s Bearing Capacity Theory 0.3 0.2 25% 0.4 Relative density, Dr 0.5 0.6 Punching shear 0.7 Local shear 0.8 General shear 20% S B Rectangular plates Circular plates 15% 10% 5% Circular plate diameter 203 mm (8 in.) 152 mm (6 in.) 102 mm (4 in.) 51 mm (2 in.) 51 305 mm (2 12 in.) Rectangular plate (width B) 0% 1.35 1.40 1.45 1.50 Dry unit weight, d Unit weight of water, w 1.55 Figure 3.4 Range of settlement of circular and rectangular plates at ultimate load (Df>B 5 0) in sand (Modified from Vesic, 1963) (From Vesic, A. B. Bearing Capacity of Deep Foundations in Sand. In Highway Research Record 39, Highway Research Board, National Research Council, Washington, D.C., 1963, Figure 29, p. 138. Reproduced with permission of the Transportation Research Board.) B J I qu Df H 45 /2 A 45 /2 F C D q Df G 45 /2 45 /2 E Soil Unit weight Cohesion c Friction angle Figure 3.5 Bearing capacity failure in soil under a rough rigid continuous (strip) foundation 137 138 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity 1. The triangular zone ACD immediately under the foundation 2. The radial shear zones ADF and CDE, with the curves DE and DF being arcs of a logarithmic spiral 3. Two triangular Rankine passive zones AFH and CEG The angles CAD and ACD are assumed to be equal to the soil friction angle fr. Note that, with the replacement of the soil above the bottom of the foundation by an equivalent surcharge q, the shear resistance of the soil along the failure surfaces GI and HJ was neglected. Using equilibrium analysis, Terzaghi expressed the ultimate bearing capacity in the form qu 5 crNc 1 qNq 1 12 gBNg (continuous or strip foundation) (3.3) where cr 5 cohesion of soil g 5 unit weight of soil q 5 gDf Nc, Nq, Ng 5 bearing capacity factors that are nondimensional and are functions only of the soil friction angle fr The bearing capacity factors Nc, Nq, and Ng are defined by Nc 5 cot fr e2 (3p>42fr>2)tan fr C fr p 2 cos ¢ 1 ≤ 4 2 2 Nq 5 2 1 5 cot fr(Nq 2 1) S e2 (3p>42fr>2)tan fr fr 2 cos ¢ 45 1 ≤ 2 (3.4) (3.5) 2 and 1 Kpg Ng 5 ¢ 2 1≤ tan fr 2 cos2 fr (3.6) where Kpg 5 passive pressure coefficient. The variations of the bearing capacity factors defined by Eqs. (3.4), (3.5), and (3.6) are given in Table 3.1. To estimate the ultimate bearing capacity of square and circular foundations, Eq. (3.1) may be respectively modified to qu 5 1.3crNc 1 qNq 1 0.4gBNg (square foundation) (3.7) 3.3 Terzaghi’s Bearing Capacity Theory 139 Table 3.1 Terzaghi’s Bearing Capacity Factors—Eqs. (3.4), (3.5), and (3.6) a From Kumbhojkar (1993) a f9 Nc Nq N ga f9 Nc Nq 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 5.70 6.00 6.30 6.62 6.97 7.34 7.73 8.15 8.60 9.09 9.61 10.16 10.76 11.41 12.11 12.86 13.68 14.60 15.12 16.56 17.69 18.92 20.27 21.75 23.36 25.13 1.00 1.10 1.22 1.35 1.49 1.64 1.81 2.00 2.21 2.44 2.69 2.98 3.29 3.63 4.02 4.45 4.92 5.45 6.04 6.70 7.44 8.26 9.19 10.23 11.40 12.72 0.00 0.01 0.04 0.06 0.10 0.14 0.20 0.27 0.35 0.44 0.56 0.69 0.85 1.04 1.26 1.52 1.82 2.18 2.59 3.07 3.64 4.31 5.09 6.00 7.08 8.34 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 27.09 29.24 31.61 34.24 37.16 40.41 44.04 48.09 52.64 57.75 63.53 70.01 77.50 85.97 95.66 106.81 119.67 134.58 151.95 172.28 196.22 224.55 258.28 298.71 347.50 14.21 15.90 17.81 19.98 22.46 25.28 28.52 32.23 36.50 41.44 47.16 53.80 61.55 70.61 81.27 93.85 108.75 126.50 147.74 173.28 204.19 241.80 287.85 344.63 415.14 N ga 9.84 11.60 13.70 16.18 19.13 22.65 26.87 31.94 38.04 45.41 54.36 65.27 78.61 95.03 115.31 140.51 171.99 211.56 261.60 325.34 407.11 512.84 650.67 831.99 1072.80 From Kumbhojkar (1993) and qu 5 1.3crNc 1 qNq 1 0.3gBNg (circular foundation) (3.8) In Eq. (3.7), B equals the dimension of each side of the foundation; in Eq. (3.8), B equals the diameter of the foundation. For foundations that exhibit the local shear failure mode in soils, Terzaghi suggested the following modifications to Eqs. (3.3), (3.7), and (3.8): qu 5 23crN cr 1 qN qr 1 12gBN gr (strip foundation) qu 5 0.867crN cr 1 qN qr 1 0.4gBN gr (square foundation) (3.10) qu 5 0.867crN cr 1 qN qr 1 0.3gBN gr (circular foundation) (3.11) (3.9) 140 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity Table 3.2 Terzaghi’s Modified Bearing Capacity Factors Ncr, Nqr, and Ngr f9 N 9c N 9q N 9g f9 N 9c N 9q N 9g 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 5.70 5.90 6.10 6.30 6.51 6.74 6.97 7.22 7.47 7.74 8.02 8.32 8.63 8.96 9.31 9.67 10.06 10.47 10.90 11.36 11.85 12.37 12.92 13.51 14.14 14.80 1.00 1.07 1.14 1.22 1.30 1.39 1.49 1.59 1.70 1.82 1.94 2.08 2.22 2.38 2.55 2.73 2.92 3.13 3.36 3.61 3.88 4.17 4.48 4.82 5.20 5.60 0.00 0.005 0.02 0.04 0.055 0.074 0.10 0.128 0.16 0.20 0.24 0.30 0.35 0.42 0.48 0.57 0.67 0.76 0.88 1.03 1.12 1.35 1.55 1.74 1.97 2.25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 15.53 16.30 17.13 18.03 18.99 20.03 21.16 22.39 23.72 25.18 26.77 28.51 30.43 32.53 34.87 37.45 40.33 43.54 47.13 51.17 55.73 60.91 66.80 73.55 81.31 6.05 6.54 7.07 7.66 8.31 9.03 9.82 10.69 11.67 12.75 13.97 15.32 16.85 18.56 20.50 22.70 25.21 28.06 31.34 35.11 39.48 44.45 50.46 57.41 65.60 2.59 2.88 3.29 3.76 4.39 4.83 5.51 6.32 7.22 8.35 9.41 10.90 12.75 14.71 17.22 19.75 22.50 26.25 30.40 36.00 41.70 49.30 59.25 71.45 85.75 Ncr, Nqr, and Ngr, the modified bearing capacity factors, can be calculated by using the bearing capacity factor equations (for Nc, Nq, and Ng, respectively) by replacing fr by fr 5 tan21 ( 23 tan fr). The variation of Ncr, Nqr, and Ngr with the soil friction angle fr is given in Table 3.2. Terzaghi’s bearing capacity equations have now been modified to take into account the effects of the foundation shape (B>L), depth of embedment (Df ), and the load inclination. This is given in Section 3.6. Many design engineers, however, still use Terzaghi’s equation, which provides fairly good results considering the uncertainty of the soil conditions at various sites. 3.4 Factor of Safety Calculating the gross allowable load-bearing capacity of shallow foundations requires the application of a factor of safety (FS) to the gross ultimate bearing capacity, or qall 5 qu FS (3.12) 3.4 Factor of Safety 141 However, some practicing engineers prefer to use a factor of safety such that Net stress increase on soil 5 net ultimate bearing capacity FS (3.13) The net ultimate bearing capacity is defined as the ultimate pressure per unit area of the foundation that can be supported by the soil in excess of the pressure caused by the surrounding soil at the foundation level. If the difference between the unit weight of concrete used in the foundation and the unit weight of soil surrounding is assumed to be negligible, then qnet(u) 5 qu 2 q (3.14) where qnet(u) 5 net ultimate bearing capacity q 5 gDf So qall(net) 5 qu 2 q FS (3.15) The factor of safety as defined by Eq. (3.15) should be at least 3 in all cases. Example 3.1 A square foundation is 2 m 3 2 m in plan. The soil supporting the foundation has a friction angle of fr 5 25° and cr 5 20 kN>m2. The unit weight of soil, g, is 16.5 kN>m3. Determine the allowable gross load on the foundation with a factor of safety (FS) of 3. Assume that the depth of the foundation (Df ) is 1.5 m and that general shear failure occurs in the soil. Solution From Eq. (3.7) qu 5 1.3crNc 1 qNq 1 0.4gBNg From Table 3.1, for fr 5 25°, Nc 5 25.13 Nq 5 12.72 Ng 5 8.34 Thus, qu 5 (1.3) (20) (25.13) 1 (1.5 3 16.5) (12.72) 1 (0.4) (16.5) (2) (8.34) 5 653.38 1 314.82 1 110.09 5 1078.29 kN>m2 142 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity So, the allowable load per unit area of the foundation is qu 1078.29 5 < 359.5 kN>m2 qall 5 FS 3 Thus, the total allowable gross load is Q 5 (359.5) B2 5 (359.5) (2 3 2) 5 1438 kN 3.5 ■ Modification of Bearing Capacity Equations for Water Table Equations (3.3) and (3.7) through (3.11) give the ultimate bearing capacity, based on the assumption that the water table is located well below the foundation. However, if the water table is close to the foundation, some modifications of the bearing capacity equations will be necessary. (See Figure 3.6.) Case I. If the water table is located so that 0 # D1 # Df, the factor q in the bearing capacity equations takes the form (3.16) q 5 effective surcharge 5 D1g 1 D2 (gsat 2 gw ) where gsat 5 saturated unit weight of soil gw 5 unit weight of water Also, the value of g in the last term of the equations has to be replaced by gr 5 gsat 2 gw. Case II. For a water table located so that 0 # d # B, (3.17) q 5 gDf In this case, the factor g in the last term of the bearing capacity equations must be replaced by the factor g 5 gr 1 Groundwater table d (g 2 gr) B (3.18) D1 Case I Df D2 B d Groundwater table Case II sat saturated unit weight Figure 3.6 Modification of bearing capacity equations for water table 3.6 The General Bearing Capacity Equation 143 The preceding modifications are based on the assumption that there is no seepage force in the soil. Case III. When the water table is located so that d $ B, the water will have no effect on the ultimate bearing capacity. 3.6 The General Bearing Capacity Equation The ultimate bearing capacity equations (3.3), (3.7), and (3.8) are for continuous, square, and circular foundations only; they do not address the case of rectangular foundations (0 , B>L , 1). Also, the equations do not take into account the shearing resistance along the failure surface in soil above the bottom of the foundation (the portion of the failure surface marked as GI and HJ in Figure 3.5). In addition, the load on the foundation may be inclined. To account for all these shortcomings, Meyerhof (1963) suggested the following form of the general bearing capacity equation: qu 5 crNcFcsFcdFci 1 qNqFqsFqdFqi 1 12 gBNgFgsFgdFgi (3.19) In this equation: cr 5 q5 g5 B5 Fcs, Fqs, Fgs 5 Fcd, Fqd, Fgd 5 Fci, Fqi, Fgi 5 Nc, Nq, Ng 5 cohesion effective stress at the level of the bottom of the foundation unit weight of soil width of foundation (5 diameter for a circular foundation) shape factors depth factors load inclination factors bearing capacity factors The equations for determining the various factors given in Eq. (3.19) are described briefly in the sections that follow. Note that the original equation for ultimate bearing capacity is derived only for the plane-strain case (i.e., for continuous foundations). The shape, depth, and load inclination factors are empirical factors based on experimental data. Bearing Capacity Factors The basic nature of the failure surface in soil suggested by Terzaghi now appears to have been borne out by laboratory and field studies of bearing capacity (Vesic, 1973). However, the angle a shown in Figure 3.5 is closer to 45 1 fr>2 than to fr. If this change is accepted, the values of Nc, Nq, and Ng for a given soil friction angle will also change from those given in Table 3.1. With a 5 45 1 fr>2, it can be shown that Nq 5 tan2 ¢45 1 fr p tan fr ≤e 2 (3.20) 144 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity and Nc 5 (Nq 2 1) cot fr (3.21) Equation (3.21) for Nc was originally derived by Prandtl (1921), and Eq. (3.20) for Nq was presented by Reissner (1924). Caquot and Kerisel (1953) and Vesic (1973) gave the relation for Ng as Ng 5 2 (Nq 1 1) tan fr (3.22) Table 3.3 shows the variation of the preceding bearing capacity factors with soil friction angles. Table 3.3 Bearing Capacity Factors f9 Nc Nq Ng f9 Nc Nq Ng 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 5.14 5.38 5.63 5.90 6.19 6.49 6.81 7.16 7.53 7.92 8.35 8.80 9.28 9.81 10.37 10.98 11.63 12.34 13.10 13.93 14.83 15.82 16.88 18.05 19.32 20.72 1.00 1.09 1.20 1.31 1.43 1.57 1.72 1.88 2.06 2.25 2.47 2.71 2.97 3.26 3.59 3.94 4.34 4.77 5.26 5.80 6.40 7.07 7.82 8.66 9.60 10.66 0.00 0.07 0.15 0.24 0.34 0.45 0.57 0.71 0.86 1.03 1.22 1.44 1.69 1.97 2.29 2.65 3.06 3.53 4.07 4.68 5.39 6.20 7.13 8.20 9.44 10.88 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 22.25 23.94 25.80 27.86 30.14 32.67 35.49 38.64 42.16 46.12 50.59 55.63 61.35 67.87 75.31 83.86 93.71 105.11 118.37 133.88 152.10 173.64 199.26 229.93 266.89 11.85 13.20 14.72 16.44 18.40 20.63 23.18 26.09 29.44 33.30 37.75 42.92 48.93 55.96 64.20 73.90 85.38 99.02 115.31 134.88 158.51 187.21 222.31 265.51 319.07 12.54 14.47 16.72 19.34 22.40 25.99 30.22 35.19 41.06 48.03 56.31 66.19 78.03 92.25 109.41 130.22 155.55 186.54 224.64 271.76 330.35 403.67 496.01 613.16 762.89 Shape, Depth, Inclination Factors Commonly used shape, depth, and inclination factors are given in Table 3.4. 3.6 The General Bearing Capacity Equation 145 Table 3.4 Shape, Depth and Inclination Factors (DeBeer (1970); Hansen (1970); Meyerhof (1963); Meyerhof and Hanna (1981)) Factor Shape Depth Relationship Reference B Nq Fcs 5 1 1 a b a b L Nc B Fqs 5 1 1 a b tan fr L B Fgs 5 120.4 a b L Df 1 B For ⫽ 0: Df Fcd 5 1 1 0.4 a b B DeBeer (1970) Hansen (1970) Fqd ⫽ 1 F␥d ⫽ 1 For ⬘ ⬎ 0: Fcd 5 Fqd 2 1 2 Fqd Nc tan fr Fqd 5 1 1 2 tan fr (1 2 sin fr) 2 a Df B b F␥d ⫽ 1 Df B 1 For ⫽ 0: Df Fcd 5 1 1 0.4 tan 21 a b B ('')''* radians Fqd ⫽ 1 F␥d ⫽ 1 For ⬘ ⬎ 0: Fcd 5 Fqd 2 1 2 Fqd Nc tan f9 Df Fqd 5 1 1 2 tan fr (1 2 sin fr) 2 tan 21 a b B ('')''* radians F␥d ⫽ 1 Inclination Fci 5 Fqi 5 a1 2 Fgi 5 a1 2 b f9 b° 2 b 90° b  ⫽ inclination of the load on the foundation with respect to the vertical Meyerhof (1963); Hanna and Meyerhof (1981) 146 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity Example 3.2 Solve Example Problem 3.1 using Eq. (3.19). Solution From Eq. (3.19), qu 5 c9NcFcsFcdFci 1 qNqFqsFqdFqt 1 1 gBNgFgsFgdFgt 2 Since the load is vertical, Fci ⫽ Fqi ⫽ F␥i ⫽ 1. From Table 3.3 for ⬘ ⫽ 25°, Nc ⫽ 20.72, Nq ⫽ 10.66, and N␥ ⫽ 10.88. Using Table 3.4, B Nq 2 10.66 Fcs 5 1 1 a b a b 5 1 1 a b a b 5 1.514 L Nc 2 20.72 B 2 Fqs 5 1 1 a b tanf9 5 1 1 a b tan 25 5 1.466 L 2 B 2 Fgs 5 1 2 0.4a b 5 1 2 0.4a b 5 0.6 L 2 Fqd 5 1 1 2 tanfr (1 2 sinfr) 2 a Df B b 5 1 1 (2) (tan 25) (1 2 sin 25) 2 a Fcd 5 Fqd 2 1 2 Fqd Nc tanf9 5 1.233 2 c 1.5 b 5 1.233 2 1 2 1.233 d 5 1.257 (20.72) (tan 25) Fgd 5 1 Hence, qu ⫽ (20)(20.72)(1.514)(1.257)(1) ⫹(1.5 ⫻ 16.5)(10.66)(1.466)(1.233)(1) 1 1 (16.5) (2) (10.88) (0.6) (1) (1) 2 ⫽ 788.6 ⫹ 476.9 ⫹ 107.7 ⫽ 1373.2 kN/m2 qall 5 qu 1373.2 5 5 457.7 kN>m2 FS 3 Q ⫽ (457.7)(2 ⫻ 2) ⫽ 1830.8 kN ■ 3.6 The General Bearing Capacity Equation 147 Example 3.3 A square foundation (B 3 B) has to be constructed as shown in Figure 3.7. Assume that g 5 16.5 kN>m3, gsat 5 18.55 kN>m3, fr 5 34°, Df 5 1.22 m, and D1 5 0.61 m. The gross allowable load, Qall, with FS 5 3 is 667.2 kN. Determine the size of the footing. Use Eq. (3.19). D1 Water table ; ; c 0 sat c 0 Df BB Figure 3.7 A square foundation Solution We have qall 5 Qall B2 5 667.2 kN>m2 B2 (a) From Eq. (3.19) (with cr 5 0), for vertical loading, we obtain qall 5 qu 1 1 5 ¢ qNqFqsFqd 1 grBNgFgsFgd ≤ FS 3 2 For fr 5 34°, from Table 3.3, Nq 5 29.44 and Ng 5 41.06. Hence, Fqs 5 1 1 B tan fr 5 1 1 tan 34 5 1.67 L Fgs 5 1 2 0.4 ¢ B ≤ 5 1 2 0.4 5 0.6 L Fqd 5 1 1 2 tan fr (1 2 sin fr) 2 Df B 5 1 1 2 tan 34 (1 2 sin 34) 2 4 1.05 511 B B Fgd 5 1 and q 5 (0.61) (16.5) 1 0.61 (18.55 2 9.81) 5 15.4 kN>m2 148 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity So qall 5 1 1.05 B(15.4) (29.44) (1.67) ¢1 1 ≤ 3 B 1 1 ¢ ≤ (18.55 2 9.81) (B) (41.06) (0.6) (1) R 2 5 252.38 1 (b) 265 1 35.89B B Combining Eqs. (a) and (b) results in 667.2 265 5 252.38 1 1 35.89B 2 B B By trial and error, we find that B < 1.3 m. 3.7 ■ Case Studies on Ultimate Bearing Capacity In this section, we will consider two field observations related to the ultimate bearing capacity of foundations on soft clay. The failure loads on the foundations in the field will be compared with those estimated from the theory presented in Section 3.6. Foundation Failure of a Concrete Silo An excellent case of bearing capacity failure of a 6-m diameter concrete silo was provided by Bozozuk (1972). The concrete tower silo was 21 m high and was constructed over soft clay on a ring foundation. Figure 3.8 shows the variation of the undrained shear strength (cu) obtained from field vane shear tests at the site. The groundwater table was located at about 0.6 m below the ground surface. On September 30, 1970, just after it was filled to capacity for the first time with corn silage, the concrete tower silo suddenly overturned due to bearing capacity failure. Figure 3.9 shows the approximate profile of the failure surface in soil. The failure surface extended to about 7 m below the ground surface. Bozozuk (1972) provided the following average parameters for the soil in the failure zone and the foundation: • • • • Load per unit area on the foundation when failure occurred < 160 kN>m2 Average plasticity index of clay (PI) < 36 Average undrained shear strength (cu) from 0.6 to 7 m depth obtained from field vane shear tests < 27.1 kN>m2 From Figure 3.9, B < 7.2 m and Df < 1.52 m. cu (kN/m2) 40 60 20 0 80 100 1 Depth (m) 2 3 4 5 Figure 3.8 Variation of cu with depth obtained from field vane shear test 6 50 Original position of foundation 1.46 m 0 22 2 1m m 30 0.9 4 6 Paved apron Original ground surface 7.2 Depth below paved apron (m) 1 2 Collapsed silo 50 Upheaval 22 45 m 1. 60 8 10 12 Figure 3.9 Approximate profile of silo failure (Adapted from Bozozuk, 1972) 149 150 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity We can now calculate the factor of safety against bearing capacity failure. From Eq. (3.19) qu 5 crNcFcsFcdFci 1 qNcFqsFqdFqi 1 12 gB NgFgsFgdFgi For f 5 0 condition and vertical loading, cr 5 cu, Nc 5 5.14, Nq 5 1, Ng 5 0, and Fci 5 Fqi 5 Fgi 5 0. Also, from Table 3.4, Fcs 5 1 1 a 7.2 1 ba b 5 1.195 7.2 5.14 Fqs 5 1 Fcd 5 1 1 (0.4) a 1.52 b 5 1.08 7.2 Fqd 5 1 Thus, qu 5 (cu ) (5.14) (1.195) (1.08) (1) 1 (g) (1.52) Assuming g < 18 kN>m3, qu 5 6.63cu 1 27.36 (3.23) According to Eqs. (2.34) and (2.35a), cu(corrected) 5 l cu(VST) l 5 1.7 2 0.54 log 3PI(%)4 For this case, PI < 36 and cu(VST) 5 27.1 kN>m2. So cu(corrected) 5 51.7 2 0.54 log 3PI(%)46cu(VST) 5 (1.7 2 0.54 log 36) (27.1) < 23.3 kN>m2 Substituting this value of cu in Eq. (3.23) qu 5 (6.63) (23.3) 1 27.36 5 181.8 kN>m2 The factor of safety against bearing capacity failure FS 5 qu 181.8 5 5 1.14 applied load per unit area 160 This factor of safety is too low and approximately equals one, for which the failure occurred. Load Tests on Small Foundations in Soft Bangkok Clay Brand et al. (1972) reported load test results for five small square foundations in soft Bangkok clay in Rangsit, Thailand. The foundations were 0.6 m ⫻ 0.6 m, 0.675 m ⫻ 0.675 m, 0.75 m ⫻ 0.75 m, 0.9 m ⫻ 0.9 m, and 1.05 m ⫻ 1.05 m. The depth of of the foundations (Df) was 1.5 m in all cases. Figure 3.10 shows the vane shear test results for clay. Based on the variation of cu(VST) with depth, it can be approximated that cu(VST) is about 35 kN/m2 for depths between zero to 1.5 m measured from the ground surface, and cu(VST) is approximately equal to 24 kN/m2 for depths varying from 1.5 to 8 m. Other properties of the clay are • • • Liquid limit ⫽ 80 Plastic limit ⫽ 40 Sensitivity ⬇ 5 3.7 Case Studies on Ultimate Bearing Capacity 151 cu (VST) (kN/m2) 0 10 20 30 40 1 2 Depth (m) 3 4 5 6 7 8 Figure 3.10 Variation of cu(VST) with depth for soft Bangkok clay Figure 3.11 shows the load-settlement plots obtained from the bearing-capacity tests on all five foundations. The ultimate loads, Qu, obtained from each test are shown in Figure 3.11 and given in Table 3.5. The ultimate load is defined as the point where the load-settlement plot becomes practically linear. From Eq. (3.19), qu 5 c9NcFcsFcdFci 1 qNqFqsFqdFqi 1 1 gBNgFgsFgdFgi 2 For undrained condition and vertical loading (that is, ⫽ 0) from Tables 3.3 and 3.4, • • • • • • Fci ⫽ Fqi ⫽ F␥i ⫽ 1 c⬘ ⫽ cu, Nc ⫽ 5.14, Nq ⫽ 1, and N␥ ⫽ 0 B Nq 1 Fcs 5 1 1 a b a b 5 1 1 (1) a b 5 1.195 L Nc 5.14 Fqs ⫽ 1 Fqd ⫽ 1 Df 1.5 Fcd 5 1 1 0.4 tan 21 a b 5 1 1 0.4 tan 21 a b B B (3.24) 152 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity Load (kN) 0 40 0 200 160 120 80 Qu (ultimate load) Settlement (mm) 10 20 B = 0.675 m 30 B = 1.05 m B = 0.6 m B = 0.75 m B = 0.9 m 40 Figure 3.11 Load-settlement plots obtained from bearing capacity tests (Note: Df /B ⬎ 1 in all cases) Thus, qu ⫽ (5.14)(cu)(1.195)Fcd ⫹ q (3.25) The values of cu(VST) need to be corrected for use in Eq. (3.25). From Eq. (2.34), cu ⫽ cu(VST) From Eq. (2.35b), ⫽ 1.18e⫺0.08(PI) ⫹ 0.57 ⫽ 1.18e⫺0.08(80 ⫺ 40) ⫹ 0.57 ⫽ 0.62 From Eq. (2.35c), ⫽ 7.01e⫺0.08(LL) ⫹ 0.57 ⫽ 7.01e⫺0.08(80) ⫹ 0.57 ⫽ 0.58 Table 3.5 Comparison of Ultimate Bearing Capacity—Theory versus Field Test Results B (m) (1) Df (m) (2) 0.600 0.675 0.750 0.900 1.050 1.5 1.5 1.5 1.5 1.5 Fcd (3) qu(theory)‡‡ (kN/m2) (4) 1.476 1.459 1.443 1.412 1.384 158.3 156.8 155.4 152.6 150.16 ‡ Qu(field) (kN) (5) qu(field)‡‡‡ (kN/m2) (6) qu(field) 2 qu(theory) (%) qu(field) 60 71 90 124 140 166.6 155.8 160.6 153.0 127.0 4.98 ⫺0.64 2.87 0.27 ⫺18.24 Eq. (3.24); ‡‡Eq. (3.26); ‡‡‡Qu(field)/B2 ⫽ qu(field) ‡ (7) 3.8 Effect of Soil Compressibility 153 So the average value of ⬇ 0.6. Hence, cu ⫽ cu(VST) ⫽ (0.6)(24) ⫽ 14.4 kN/m2 Let us assume ␥ ⫽ 18.5 kN/m2. So q ⫽ ␥Df ⫽ (18.5)(1.5) ⫽ 27.75 kN/m2 Substituting cu ⫽ 14.4 kN/m2 and q ⫽ 27.75 kN/m2 into Eq. (3.25), we obtain qu(kN/m2) ⫽ 88.4Fcd ⫹ 27.75 (3.26) The values of qu calculated using Eq. (3.26) are given in column 4 of Table 3.5. Also, the qu determined from the field tests are given in column 6. The theoretical and field values of qu compare very well. The important lessons learned from this study are 1. The ultimate bearing capacity is a function of cu. If Eq. (2.35a) would have been used to correct the undrained shear strength, the theoretical values of qu would have varied between 200 kN/m2 and 210 kN/m2. These values are about 25% to 55% more than those obtained from the field and are on the unsafe side. 2. It is important to recognize that empirical correlations like those given in Eqs. (2.35a), (2.35b) and (2.35c) are sometimes site specific. Thus, proper engineering judgment and any record of past studies would be helpful in the evaluation of bearing capacity. 3.8 Effect of Soil Compressibility In Section 3.3, Eqs. (3.3), (3.7), and (3.8), which apply to the case of general shear failure, were modified to Eqs. (3.9), (3.10), and (3.11) to take into account the change of failure mode in soil (i.e., local shear failure). The change of failure mode is due to soil compressibility, to account for which Vesic (1973) proposed the following modification of Eq. (3.19): qu 5 crNcFcsFcdFcc 1 qNqFqsFqdFqc 1 12 gBNgFgsFgdFgc (3.27) In this equation, Fcc, Fqc, and Fgc are soil compressibility factors. The soil compressibility factors were derived by Vesic (1973) by analogy to the expansion of cavities. According to that theory, in order to calculate Fcc, Fqc, and Fgc, the following steps should be taken: Step 1. Calculate the rigidity index, Ir, of the soil at a depth approximately B>2 below the bottom of the foundation, or Ir 5 Gs cr 1 qr tan fr where Gs 5 shear modulus of the soil q 5 effective overburden pressure at a depth of Df 1 B>2 (3.28) 154 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity Step 2. The critical rigidity index, Ir(cr), can be expressed as fr 1 B Ir(cr) 5 bexp B ¢ 3.30 2 0.45 ≤ cot ¢ 45 2 ≤ R r 2 L 2 Step 3. (3.29) The variations of Ir(cr) with B>L are given in Table 3.6. If Ir $ Ir(cr), then Fcc 5 Fqc 5 Fgc 5 1 However, if Ir , Ir(cr), then Fgc 5 Fqc 5 exp b ¢ 24.4 1 0.6 (3.07 sin fr) (log 2Ir ) B ≤ tan fr 1 B Rr L 1 1 sin fr (3.30) Figure 3.12 shows the variation of Fgc 5 Fqc [see Eq. (3.30)] with fr and Ir. For f 5 0, Fcc 5 0.32 1 0.12 B 1 0.60 log Ir L (3.31) For fr . 0, Fcc 5 Fqc 2 1 2 Fqc (3.32) Nq tan fr Table 3.6 Variation of Ir(cr) with ⬘ and B/L Ir(cr) ⬘ (deg) B/L ⴝ 0 B/L ⴝ 0.2 B/L ⴝ 0.4 B/L ⴝ 0.6 B/L ⴝ 0.8 B/L ⴝ 1.0 0 5 10 15 20 25 30 35 40 45 13.56 18.30 25.53 36.85 55.66 88.93 151.78 283.20 593.09 1440.94 12.39 16.59 22.93 32.77 48.95 77.21 129.88 238.24 488.97 1159.56 11.32 15.04 20.60 29.14 43.04 67.04 111.13 200.41 403.13 933.19 10.35 13.63 18.50 25.92 37.85 58.20 95.09 168.59 332.35 750.90 9.46 12.36 16.62 23.05 33.29 50.53 81.36 141.82 274.01 604.26 8.64 11.20 14.93 20.49 29.27 43.88 69.62 119.31 225.90 486.26 3.8 Effect of Soil Compressibility 1.0 1.0 500 100 250 0.8 250 0.8 50 50 0.6 10 0.4 100 F c Fqc F c Fqc 155 25 5 2.5 0.2 0.6 25 10 0.4 5 2.5 0.2 Ir 1 0 500 Ir 1 0 0 10 20 30 50 40 Soil friction angle, (deg) L (a) 1 B 0 10 20 30 50 40 Soil friction angle, (deg) L (b) 5 B Figure 3.12 Variation of Fgc 5 Fqc with Ir and fr Example 3.4 For a shallow foundation, B 5 0.6 m, L 5 1.2 m, and Df 5 0.6 m. The known soil characteristics are as follows: Soil: fr 5 25° cr 5 48 kN>m2 g 5 18 kN>m3 Modulus of elasticity, Es 5 620 kN>m2 Poisson’s ratio, ms 5 0.3 Calculate the ultimate bearing capacity. Solution From Eq. (3.28), Ir 5 Gs cr 1 qr tan fr However, Gs 5 Es 2 (1 1 ms ) So Ir 5 Now, qr 5 g ¢ Df 1 Es 2 (1 1 ms ) 3cr 1 qr tan fr4 B 0.6 ≤ 5 18 ¢0.6 1 ≤ 5 16.2 kN>m2 2 2 156 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity Thus, Ir 5 From Eq. (3.29), 620 5 4.29 2 (1 1 0.3) 348 1 16.2 tan 254 fr 1 B Ir(cr) 5 bexp B ¢3.3 2 0.45 ≤ cot ¢45 2 ≤ R r 2 L 2 1 25 0.6 5 bexp B ¢ 3.3 2 0.45 ≤ cot ¢45 2 ≤ R r 5 62.41 2 1.2 2 Since Ir(cr) . Ir, we use Eqs. (3.30) and (3.32) to obtain Fgc 5 Fqc 5 exp b ¢ 24.4 1 0.6 5 exp b ¢ 24.4 1 0.6 1 c (3.07 sin fr)log(2Ir ) B ≤ tan fr 1 B Rr L 1 1 sin fr 0.6 ≤ tan 25 1.2 (3.07 sin 25)log(2 3 4.29) R r 5 0.347 1 1 sin 25 and Fcc 5 Fqc 2 1 2 Fqc Nc tan fr For fr 5 25°, Nc 5 20.72 (see Table 3.3); therefore, 1 2 0.347 Fcc 5 0.347 2 5 0.279 20.72 tan 25 Now, from Eq. (3.27), qu 5 crNcFcsFcdFcc 1 qNqFqsFqdFqc 1 12gBNgFgsFgdFgc From Table 3.3, for fr 5 25°, Nc 5 20.72, Nq 5 10.66, and Ng 5 10.88. Consequently, Nq B 10.66 0.6 Fcs 5 1 1 ¢ ≤ ¢ ≤ 5 1 1 ¢ ≤¢ ≤ 5 1.257 Nc L 20.72 1.2 Fqs 5 1 1 B 0.6 tan fr 5 1 1 tan 25 5 1.233 L 1.2 Fgs 5 1 2 0.4 ¢ B 0.6 ≤ 5 1 2 0.4 5 0.8 L 1.2 Fqd 5 1 1 2 tan fr (1 2 sin fr) 2 ¢ Df 5 1 1 2 tan 25 (1 2 sin 25) 2 ¢ Fcd 5 Fqd 2 1 2 Fqd Nc tan fr 5 1.311 2 B ≤ 0.6 ≤ 5 1.311 0.6 1 2 1.311 5 1.343 20.72 tan 25 3.9 Eccentrically Loaded Foundations 157 and Fgd 5 1 Thus, qu 5 (48) (20.72) (1.257) (1.343) (0.279) 1 (0.6 3 18) (10.66) (1.233) (1.311) (0.347)1( 12 ) (18) (0.6) (10.88) (0.8) (1) (0.347) 5 549.32 kN , m2 3.9 ■ Eccentrically Loaded Foundations In several instances, as with the base of a retaining wall, foundations are subjected to moments in addition to the vertical load, as shown in Figure 3.13a. In such cases, the distribution of pressure by the foundation on the soil is not uniform. The nominal distribution of pressure is qmax 5 Q 6M 1 2 BL BL (3.33) qmin 5 Q 6M 2 2 BL BL (3.34) and Q Q e M B B BL For e < B/6 qmin qmax L For e > B/6 qmax (a) Figure 3.13 Eccentrically loaded foundations 2e B (b) 158 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity where Q 5 total vertical load M 5 moment on the foundation Figure 3.13b shows a force system equivalent to that shown in Figure 3.13a. The distance e5 M Q (3.35) is the eccentricity. Substituting Eq. (3.35) into Eqs. (3.33) and (3.34) gives qmax 5 Q 6e ¢1 1 ≤ BL B (3.36) qmin 5 Q 6e ¢1 2 ≤ BL B (3.37) and Note that, in these equations, when the eccentricity e becomes B>6, qmin is zero. For e . B>6, qmin will be negative, which means that tension will develop. Because soil cannot take any tension, there will then be a separation between the foundation and the soil underlying it. The nature of the pressure distribution on the soil will be as shown in Figure 3.13a. The value of qmax is then qmax 5 4Q 3L(B 2 2e) (3.38) The exact distribution of pressure is difficult to estimate. Figure 3.14 shows the nature of failure surface in soil for a surface strip foundation subjected to an eccentric load. The factor of safety for such type of loading against bearing capacity failure can be evaluated as FS 5 Qult Q (3.39) where Qult 5 ultimate load-carrying capacity. The following sections describe several theories for determining Qult. Qult e B Figure 3.14 Nature of failure surface in soil supporting a strip foundation subjected to eccentric loading (Note: Df 5 0; Qult is ultimate load per unit length of foundation) 3.10 Ultimate Bearing Capacity under Eccentric Loading—One-Way Eccentricity 3.10 159 Ultimate Bearing Capacity under Eccentric Loading—One-Way Eccentricity Effective Area Method (Meyerhoff, 1953) In 1953, Meyerhof proposed a theory that is generally referred to as the effective area method. The following is a step-by-step procedure for determining the ultimate load that the soil can support and the factor of safety against bearing capacity failure: Step 1. Determine the effective dimensions of the foundation (Figure 3.13b): Br 5 effective width 5 B 2 2e Lr 5 effective length 5 L Note that if the eccentricity were in the direction of the length of the foundation, the value of Lr would be equal to L 2 2e. The value of Br would equal B. The smaller of the two dimensions (i.e., Lr and Br) is the effective width of the foundation. Step 2. Use Eq. (3.19) for the ultimate bearing capacity: qur 5 crNcFcsFcdFci 1 qNqFqsFqdFqi 1 12 gBrNgFgsFgdFgi (3.40) To evaluate Fcs, Fqs, and Fgs, use the relationships given in Table 3.4 with effective length and effective width dimensions instead of L and B, respectively. To determine Fcd, Fqd, and Fgd, use the relationships given in Table 3.4. However, do not replace B with Br. Step 3. The total ultimate load that the foundation can sustain is Ar & $'%' Qult 5 (3.41) q ru (Br) (Lr) where Ar 5 effective area. Step 4. The factor of safety against bearing capacity failure is Qult FS 5 Q Prakash and Saran Theory Prakash and Saran (1971) analyzed the problem of ultimate bearing capacity of eccentrically and vertically loaded continuous (strip) foundations by using the one-sided failure surface in soil, as shown in Figure 3.14. According to this theory, the ultimate load per unit length of a continuous foundation can be estimated as 1 Q ult 5 B ccrNc(e) 1 qNq(e) 1 gBNg(e) d 2 (3.42) where Nc(e), Nq(e), N␥(e) ⫽ bearing capacity factors under eccentric loading. The variations of Nc(e), Nq(e), and N␥(e) with soil friction angle ⬘ are given in Figures 3.15, 3.16, and 3.17. For rectangular foundations, the ultimate load can be given as 1 Q ult 5 BL ccrNc(e)Fcs(e) 1 qNq(e)Fqs(e) 1 gBNg(e)Fgs(e) d 2 where Fcs(e), Fqs(e), and F␥s(e) ⫽ shape factors. (3.43) 160 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity 60 40 e/B = 0 Nc (e) 0.1 0.2 20 0.3 0.4 f9 5 408 0 0 10 20 Friction angle, (deg) 30 40 e/B Nc(e) 0 0.1 0.2 0.3 0.4 94.83 66.60 54.45 36.3 18.15 Figure 3.15 Variation of Nc(e) with fr Prakash and Saran (1971) also recommended the following for the shape factors: Fcs(e) 5 1.2 2 0.025 L (with a minimum of 1.0) B Fqs(e) 5 1 (3.44) (3.45) and Fgs(e) 5 1.0 1 a 2e B 3 e B 2 2 0.68b 1 c0.43 2 a b a b d a b B L 2 B L (3.46) Reduction Factor Method (For Granular Soil) Purkayastha and Char (1977) carried out stability analysis of eccentrically loaded continuous foundations supported by a layer of sand using the method of slices. Based on that analysis, they proposed 3.10 Ultimate Bearing Capacity under Eccentric Loading—One-Way Eccentricity 161 60 40 e/B = 0 Nq (e) 0.1 20 0.2 0.3 f9 5 408 0.4 0 0 10 20 Friction angle, (deg) 30 40 e/B Nq(e) 0 0.1 0.2 0.3 0.4 81.27 56.09 45.18 30.18 15.06 Figure 3.16 Variation of Nq(e) with fr Rk 5 1 2 qu(eccentric) qu(centric) (3.47) where Rk ⫽ reduction factor qu(eccentric) ⫽ ultimate bearing capacity of eccentrically loaded continuous foundations qu(centric) ⫽ ultimate bearing capacity of centrally loaded continuous foundations The magnitude of Rk can be expressed as e k Rk 5 aa b B where a and k are functions of the embedment ratio Df /B (Table 3.7). (3.48) 162 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity 60 40 e/B = 0 N(e) 0.1 20 0.2 f9 5 408 0.3 0.4 0 0 10 20 Friction angle, (deg) 30 40 e,B Ng(e) 0 0.1 0.2 0.3 0.4 115.80 71.80 41.60 18.50 4.62 Figure 3.17 Variation of Ng(e) with fr Table 3.7 Variations of a and k [Eq. (3.48)] Df/B a k 0.00 0.25 0.50 1.00 1.862 1.811 1.754 1.820 0.73 0.785 0.80 0.888 Hence, combining Eqs. (3.47) and (3.48) e k qu(eccentric) 5 qu(centric) (1 2 Rk ) 5 qu(centric) c1 2 aa b d B (3.49) 1 qu(centric) 5 qNqFqd 1 gBNgFgd 2 (3.50) where 3.10 Ultimate Bearing Capacity under Eccentric Loading—One-Way Eccentricity 163 The relationships for Fqd, and F␥d are given in Table 3.4. The ultimate load per unit length of the foundation can then be given as Qu ⫽ Bqu(eccentric) (3.51) Example 3.5 A continuous foundation is shown in Figure 3.18. If the load eccentricity is 0.2 m, determine the ultimate load, Qult, per unit length of the foundation. Use Meyerhof’s effective area method. Solution For c⬘ ⫽ 0, Eq. (3.40) gives qu9 5 qNqFqsFqdFqi 1 1 grBrNgFgsFgdFgi 2 where q ⫽ (16.5) (1.5) ⫽ 24.75 kN/m2. Sand 40 c 0 16.5 kN/m3 1.5 m 2m Figure 3.18 A continuous foundation with load eccentricity For ⬘ ⫽ 40°, from Table 3.3, Nq ⫽ 64.2 and N␥ ⫽ 109.41. Also, B⬘ ⫽ 2 ⫺ (2)(0.2) ⫽ 1.6 m Because the foundation in question is a continuous foundation, B⬘/L⬘ is zero. Hence, Fqs ⫽ 1, F␥s ⫽ 1. From Table 3.4, Fqi ⫽ F␥ i ⫽ 1 Fqd 5 1 1 2 tan fr(1 2 sin fr) 2 Df B 5 1 1 0.214a 1.5 b 5 1.16 2 F␥d ⫽ 1 and qur 5 (24.75) (64.2) (1) (1.16) (1) 1 1 a b (16.5) (1.6) (109.41) (1) (1) (1) 5 3287.39 kN>m2 2 Consequently, Qult ⫽ (B⬘)(1)(qu⬘) ⫽ (1.6)(1)(3287.39) ⬇ 5260 kN ■ 164 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity Example 3.6 Solve Example 3.5 using Eq. (3.42). Solution Since c⬘ ⫽ 0 1 Qult 5 B cqNq(e) 1 gBNg(e) d 2 e 0.2 5 5 0.1 B 2 For ⬘ ⫽ 40° and e/B ⫽ 0.1, Figures 3.16 and 3.17 give Nq(e) ⫽ 56.09 and N␥ (e) ⬇ 71.8. Hence, Qult ⫽ 2[(24.75)(56.09) ⫹ (12)(16.5)(2)(71.8)] ⫽ 5146 kN ■ Example 3.7 Solve Example 3.5 using Eq. (3.49). Solution With c⬘ ⫽ 0, 1 qu(centric) 5 qNqFqd 1 gBNgFgd 2 For ⬘ ⫽ 40°, Nq ⫽ 64.2 and N␥ ⫽ 109.41 (see Table 3.3). Hence, Fqd ⫽ 1.16 and F␥d ⫽ 1 (see Example 3.5) 1 qu(centric) 5 (24.75) (64.2) (1.16) 1 (16.5) (2) (109.41) (1) 2 5 1843.18 1 1805.27 5 3648.45 kN>m2 From Eq. (3.48), e k Rk 5 aa b B For Df /B ⫽ 1.5/2 ⫽ 0.75, Table 3.7 gives a ⬇ 1.75 and k ⬇ 0.85. Hence, Rk 5 1.79a 0.2 0.85 b 5 0.253 2 Qu ⫽ Bqu(eccentric) ⫽ Bqu(centric)(1 ⫺ Rk) ⫽ (2)(3648.45)(1 ⫺ 0.253) ⬇ 5451 kN ■ 3.11 Bearing Capacity—Two-way Eccentricity 3.11 165 Bearing Capacity—Two-way Eccentricity Consider a situation in which a foundation is subjected to a vertical ultimate load Qult and a moment M, as shown in Figures 3.19a and b. For this case, the components of the moment M about the x- and y-axes can be determined as Mx and My, respectively. (See Figure 3.19.) This condition is equivalent to a load Qult placed eccentrically on the foundation with x 5 eB and y 5 eL (Figure 3.19d). Note that My eB 5 (3.52) Qult and eL 5 Mx Qult (3.53) If Qult is needed, it can be obtained from Eq. (3.41); that is, Qult 5 qur Ar where, from Eq. (3.40), qur 5 crNcFcsFcdFci 1 qNqFqsFqdFqi 1 12 gBrNgFgsFgdFgi and Ar 5 effective area 5 BrLr As before, to evaluate Fcs, Fqs, and Fgs (Table 3.4), we use the effective length Lr and effective width Br instead of L and B, respectively. To calculate Fcd, Fqd, and Fgd, we do Qult M (a) BL B y eB Mx L Qult M Qult x Qult eL My B (b) (c) (d) Figure 3.19 Analysis of foundation with two-way eccentricity 166 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity not replace B with Br. In determining the effective area Ar, effective width Br, and effective length Lr, five possible cases may arise (Highter and Anders, 1985). Case I. eL>L $ 16 and eB>B $ 16. The effective area for this condition is shown in Figure 3.20, or Ar 5 12B1L1 (3.54) where B1 5 B ¢1.5 2 3eB ≤ B (3.55) L1 5 L ¢1.5 2 3eL ≤ L (3.56) and The effective length Lr is the larger of the two dimensions B1 and L1. So the effective width is Br 5 Ar Lr (3.57) Case II. eL>L , 0.5 and 0 , eB>B , 16. The effective area for this case, shown in Figure 3.21a, is Ar 5 12 (L1 1 L2 )B (3.58) The magnitudes of L1 and L2 can be determined from Figure 3.21b. The effective width is L1 or L2 Ar (whichever is larger) (3.59) Lr 5 L1 or L2 (whichever is larger) (3.60) Br 5 The effective length is Case III. eL>L , 16 and 0 , eB>B , 0.5. The effective area, shown in Figure 3.22a, is Ar 5 12 (B1 1 B2 )L Effective area B1 eB eL Qult L1 L B Figure 3.20 Effective area for the case of eL>L > 16 and eB>B > 16 (3.61) 3.11 Bearing Capacity–Two-way Eccentricity 167 Effective area B eB L2 eL Qult L1 L (a) 0.5 eB /B 0.4 0.167 0.1 0.08 0.06 eL /L 0.3 0.2 0.2 0.4 0.6 L1 /L, L2 /L (b) 2 2 0.0 0 For obtaining L2 /L 01 0 0. 4 0.0 6 8 0.0 eB /B 0.0 0.10 0.12 0.14 0.16 0.1 0.0 4 0.0 0. 0 0.8 1 1.0 For obtaining L1 /L Figure 3.21 Effective area for the case of eL>L , 0.5 and 0 , eB>B , 16 (After Highter and Anders, 1985) (Highter, W. H. and Anders, J. C. (1985). “Dimensioning Footings Subjected to Eccentric Loads,” Journal of Geotechnical Engineering, American Society of Civil Engineers, Vol. 111, No. GT5, pp. 659–665. With permission from ASCE.) The effective width is Ar L (3.62) Lr 5 L (3.63) Br 5 The effective length is The magnitudes of B1 and B2 can be determined from Figure 3.22b. Case IV. eL>L , 16 and eB>B , 16. Figure 3.23a shows the effective area for this case. The ratio B2>B, and thus B2, can be determined by using the eL>L curves that slope upward. Similarly, the ratio L2>L, and thus L2, can be determined by using the eL>L curves that slope downward. The effective area is then Ar 5 L2B 1 12 (B 1 B2 ) (L 2 L2 ) (3.64) 168 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity B1 eB eL Qult L Effective area B2 B (a) 0.5 eL /L 0.4 0.167 0.1 0.08 0.06 eB /B 0.3 0.2 0.2 0.4 0.6 B1 /B, B2 /B (b) 2 2 0.0 0 For obtaining B2 /B 01 0 0. 4 0.0 6 8 0.0 eL /L 0.0 0.10 0.12 0.14 0.16 0.1 0.0 4 0.0 0. 0 0.8 1 1.0 For obtaining B1 /B Figure 3.22 Effective area for the case of eL>L , 16 and 0 , eB>B , 0.5 (After Highter and Anders, 1985) Highter, W. H. and Anders, J. C. (1985). “Dimensioning Footings Subjected to Eccentric Loads,” Journal of Geotechnical Engineering, American Society of Civil Engineers, Vol. 111, No. GT5, pp. 659–665. With permission from ASCE.) The effective width is Ar L (3.65) Lr 5 L (3.66) Br 5 The effective length is Case V. (Circular Foundation) In the case of circular foundations under eccentric loading (Figure 3.24a), the eccentricity is always one way. The effective area Ar and the effective width Br for a circular foundation are given in a nondimensional form in Table 3.8. Once A9 and B9 are determined, the effective length can be obtained as Lr 5 Ar Br 3.11 Bearing Capacity–Two-way Eccentricity 169 B L2 eB eL L Qult Effective area B2 (a) For obtaining B2 /B 0.16 0.14 0.12 0.20 0.10 0.15 0.08 eB /B 1 0. 0.06 0.0 0.10 0.1 4 8 0.04 0.0 6 0.05 0.04 0.02 eL /L eL/L 0.02 For obtaining L2/L 0 0 0.2 0.4 0.6 B2 /B, L2 /L (b) 0.8 eR Qult R Figure 3.24 Effective area for circular foundation 1.0 Figure 3.23 Effective area for the case of eL>L , 16 and eB>B , 16 (After Highter and Anders, 1985) (Highter, W. H. and Anders, J. C. (1985). “Dimensioning Footings Subjected to Eccentric Loads,” Journal of Geotechnical Engineering, American Society of Civil Engineers, Vol. 111, No. GT5, pp. 659–665. With permission from ASCE.) 170 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity Table 3.8 Variation of Ar>R2 and Br>R with eR>R for Circular Foundations eR9 , R A9 , R 2 B9 , R 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 2.8 2.4 2.0 1.61 1.23 0.93 0.62 0.35 0.12 0 1.85 1.32 1.2 0.80 0.67 0.50 0.37 0.23 0.12 0 Example 3.8 A square foundation is shown in Figure 3.25, with eL 5 0.3 m and eB 5 0.15 m. Assume two-way eccentricity, and determine the ultimate load, Qult. Solution We have eL 0.3 5 5 0.2 L 1.5 and eB 0.15 5 5 0.1 B 1.5 This case is similar to that shown in Figure 3.21a. From Figure 3.21b, for eL>L 5 0.2 and eB>B 5 0.1, L1 < 0.85; L L1 5 (0.85) (1.5) 5 1.275 m L2 < 0.21; L L2 5 (0.21) (1.5) 5 0.315 m and From Eq. (3.58), Ar 5 12 (L1 1 L2 )B 5 12 (1.275 1 0.315) (1.5) 5 1.193 m2 3.11 Ultimate Bearing Capacity–Two-way Eccentricity Sand 18 kN/m3 30 c 0 0.7 m 1.5 m 1.5 m eB 0.15 m 1.5 m eL 0.3 m Figure 3.25 An eccentrically loaded foundation 1.5 m From Eq. (3.60), Lr 5 L1 5 1.275 m From Eq. (3.59), Br 5 Ar 1.193 5 5 0.936 m Lr 1.275 Note from Eq. (3.40) with cr 5 0, qur 5 qNqFqsFqdFqi 1 12gBrNgFgsFgdFgi where q 5 (0.7) (18) 5 12.6 kN>m2. For fr 5 30°, from Table 3.3, Nq 5 18.4 and Ng 5 22.4. Thus from Table 3.4, Fqs 5 1 1 ¢ Br 0.936 ≤ tan fr 5 1 1 ¢ ≤ tan 30° 5 1.424 Lr 1.275 Fgs 5 1 2 0.4 ¢ Br 0.936 ≤ 5 0.706 ≤ 5 1 2 0.4 ¢ Lr 1.275 Fqd 5 1 1 2 tan fr(1 2 sin fr) 2 Df B 511 and Fgd 5 1 (0.289) (0.7) 5 1.135 1.5 171 172 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity So Qult 5 Arqur 5 Ar(qNqFqsFqd 1 12gBrNgFgsFgd ) 5 (1.193) 3(12.6) (18.4) (1.424) (1.135) 1(0.5) (18) (0.936) (22.4) (0.706) (1)4 < 606 kN ■ Example 3.9 Consider the foundation shown in Figure 3.25 with the following changes: eL ⫽ 0.18 m eB ⫽ 0.12 m For the soil, ␥ ⫽ 16.5 kN/m3 ⬘ ⫽ 25° c⬘ ⫽ 25 kN/m2 Determine the ultimate load, Qult. Solution eL 0.18 5 5 0.12; L 1.5 eB 0.12 5 5 0.08 B 1.5 This is the case shown in Figure 3.23a. From Figure 3.23b, B2 < 0.1; B L2 < 0.32 L So B2 ⫽ (0.1)(1.5) ⫽ 0.15 m L2 ⫽ (0.32)(1.5) ⫽ 0.48 m From Eq. (3.64), 1 1 Ar 5 L2B 1 (B 1 B2 ) (L 2 L2 ) 5 (0.48) (1.5) 1 (1.5 1 0.15) (1.5 2 0.48) 2 2 5 0.72 1 0.8415 5 1.5615 m2 Br 5 Ar 1.5615 5 5 1.041m L 1.5 L9 5 1.5 m From Eq. (3.40), 1 qur 5 crNcFcs Fed 1 qNqFqsFqd 1 gBrNgFgsFgd 2 3.12 Bearing Capacity of a Continuous Foundation Subjected to Eccentric Inclined Load 173 For ⬘ ⫽ 25°, Table 3.3 gives Nc ⫽ 20.72, Nq ⫽ 10.66 and N␥ ⫽ 10.88. From Table 3.4, Fcs 5 1 1 a Br Nq 1.041 10.66 ba b 5 1 1 a ba b 5 1.357 Lr Nc 1.5 20.72 Fqs 5 1 1 a Br 1.041 b tanfr 5 1 1 a b tan 25 5 1.324 Lr 1.5 Fgs 5 1 2 0.4 a Br 1.041 b 5 120.4 a b 5 0.722 Lr 1.5 Df 0.7 Fqd 5 1 1 2 tan fr(1 2 sinfr) 2 a b 5 1 1 2 tan 25(12sin 25) 2 a b 5 1.145 B 1.5 Fcd 5 Fqd 2 1 2 Fqd 9 Nc tan f 5 1.145 2 1 2 1.145 5 1.16 20.72 tan 25 Fgd 5 1 Hence, qru 5 (25) (20.72) (1.357) (1.16) 1 (16.5 3 0.7) (10.66) (1.324) (1.145) 1 1 (16.5) (1.041) (10.88) (0.722) (1) 2 5 815.39 1 186.65 1 67.46 5 1069.5 kN>m2 Qult ⫽ A⬘qu⬘ ⫽ (1069.5)(1.5615) ⫽ 1670 kN 3.12 ■ Bearing Capacity of a Continuous Foundation Subjected to Eccentric Inclined Loading The problem of ultimate bearing capacity of a continuous foundation subjected to an eccentric inclined load was studied by Saran and Agarwal (1991). If a continuous foundation is located at a depth Df below the ground surface and is subjected to an eccentric load (load eccentricity 5 e) inclined at an angle b to the vertical, the ultimate capacity can be expressed as 1 (3.67) Qult 5 B ccrNc(ei) 1 qNq(ei) 1 gBNg(ei) d 2 where Nc(ei), Nq(ei), and Ng(ei) 5 bearing capacity factors q 5 gDf The variations of the bearing capacity factors with e>B, fr, and b derived by Saran and Agarwal are given in Figures 3.26, 3.27, and 3.28. 174 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity 100 60 =0 80 50 = 10 40 Nc (ei) Nc (ei) 60 e/B = 0 40 30 0.1 0.2 20 20 0.3 e/B = 0 0.1 0.2 10 0 0 10 20 30 Soil friction angle, (deg) 0.3 40 (a) 0 0 10 20 30 Soil friction angle, (deg) 40 (b) 40 = 20 30 30 20 e/B = 0 Nc(ei) Nc (ei) = 30 0.1 20 e/B = 0 0.2 10 10 0.3 0.1 0.2 0.3 0 0 0 10 20 30 Soil friction angle, (deg) 40 (c) Figure 3.26 Variation of Nc(ei) with fr, e>B, and b 0 10 20 30 Soil friction angle, (deg) (d) 40 3.12 Bearing Capacity of a Continuous Foundation Subjected to Eccentric Inclined Load 175 50 80 = 10 40 60 =0 Nq (ei) Nq (ei) 30 40 e/B = 0 20 e/B = 0 0.2 0.1 0.2 20 0.1 10 0.3 0.3 0 0 0 10 20 30 Soil friction angle, (deg) 0 40 10 20 30 Soil friction angle, (deg) 40 (b) (a) 30 30 = 20 = 30 e/B = 0 10 Nq (ei) 20 Nq(ei) 20 0.1 10 0.2 e/B = 0 0.3 0.1 0.2 0.3 0 0 0 10 20 30 Soil friction angle, (deg) 0 40 10 20 30 Soil friction angle, (deg) 40 (d) (c) Figure 3.27 Variation of Nq(ei) with fr, e>B, and b Example 3.10 A continuous foundation is shown in Figure 3.29. Estimate the ultimate load, Qult per unit length of the foundation. Solution With cr 5 0, from Eq. (3.67), 1 Qult 5 cqNq(ei) 1 gBNg(ei) d 2 176 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity 160 =0 120 = 10 80 e/B = 0 N (ei) N (ei) 80 40 e/B = 0 40 0.1 0.1 0.2 0.2 0.3 0.3 0 0 0 10 20 30 Soil friction angle, (deg) 10 40 20 30 Soil friction angle, (deg) 40 (b) (a) 30 60 = 20 20 40 N(ei) N (ei) = 30 e/B = 0 0.1 20 e/B = 0 0.1 10 0.2 0.3 0.2 0.3 0 0 20 30 Soil friction angle, (deg) 40 30 35 Soil friction angle, (deg) 40 (d) (c) Figure 3.28 Variation of Ng(ei) with fr, e>B, and b B 5 1.5 m, q 5 Dfg 5 (1) (16) 5 16 kN>m2, e>B 5 0.15>1.5 5 0.1, and b 520°. From Figures 3.27(c) and 3.28(c), Nq(ei) 5 14.2 and Ng(ei) 5 20. Hence, Qult 5 (1.5) 3(16) (14.2) 1 ( 12 ) (16) (1.5) (20) 4 5 700.8 kN , m Problems 177 Qult 20 16 kN/m3 35 c 0 1m 0.15 m 1.5 m ■ Figure 3.29 Problems 3.1 For the following cases, determine the allowable gross vertical load-bearing capacity of the foundation. Use Terzaghi’s equation and assume general shear failure in soil. Use FS 5 4. Part a. b. c. 3.2 3.3 3.4 3.5 3.6 B 1.22 m 2m 3m Df 0.91 m 1m 2m f9 25° 30° 30° g c9 2 28.75 kN>m 0 0 Foundation type 3 17.29 kN>m 17 kN>m3 16.5 kN>m3 Continuous Continuous Square A square column foundation has to carry a gross allowable load of 1805 kN (FS 5 3). Given: Df 5 1.5 m, g 5 15.9 kN>m3, fr 5 34°, and cr 5 0. Use Terzaghi’s equation to determine the size of the foundation (B). Assume general shear failure. Use the general bearing capacity equation [Eq. (3.19)] to solve the following: a. Problem 3.1a b. Problem 3.1b c. Problem 3.1c The applied load on a shallow square foundation makes an angle of 15° with the vertical. Given: B 5 1.83 m, Df 5 0.9 m, g 5 18.08 kN>m3, fr 5 25°, and cr 5 23.96 kN>m2. Use FS 5 4 and determine the gross allowable load. Use Eq. (3.19). A column foundation (Figure P3.5) is 3 m 3 2 m in plan. Given: Df 5 1.5 m, fr 5 25°, cr 5 70 kN>m2. Using Eq. (3.19) and FS 5 3, determine the net allowable load [see Eq. (3.15)] the foundation could carry. For a square foundation that is B 3 B in plan, Df 5 2 m; vertical gross allowable load, Qall 5 3330 kN, g 5 16.5 kN>m3; fr 5 30°; cr 5 0; and FS 5 4. Determine the size of the foundation. Use Eq. (3.19). 178 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity 17 kN/m3 1m 1.5 m Groundwater level sat 19.5 kN/m3 3m2m Figure P3.5 3.7 For the design of a shallow foundation, given the following: Soil: fr 5 25° cr 5 50 kN>m2 Unit weight, g 5 17 kN>m3 Modulus of elasticity, Es 5 1020 kN>m2 Poisson’s ratio, ms 5 0.35 Foundation: L 5 1.5 m B51m Df 5 1 m Calculate the ultimate bearing capacity. Use Eq. (3.27). 3.8 An eccentrically loaded foundation is shown in Figure P3.8. Use FS of 4 and determine the maximum allowable load that the foundation can carry. Use Meyerhof’s effective area method. 3.9 Repeat Problem 3.8 using Prakash and Saran’s method. 3.10 For an eccentrically loaded continuous foundation on sand, given B ⫽ 1.8 m, Df ⫽ 0.9 m, e/B ⫽ 0.12 (one-way eccentricity), ␥ ⫽ 16 kN/m3, and r ⫽ 35°. Using the reduction factor method, estimate the ultimate load per unit length of the foundation. (Eccentricity in one direction only) 0.1 m 0.8 m Qall 17 kN/m3 c 0 32 1.5 m 1.5 m Centerline Figure P3.8 3.11 An eccentrically loaded continuous foundation is shown in Figure P3.11. Determine the ultimate load Qu per unit length that the foundation can carry. Use the reduction factor method. 3.12 A square footing is shown in Figure P3.12. Use FS 5 6, and determine the size of the footing. Use Prakash and Saran theory [Eq. (3.43)]. References 179 Qu 16.5 kN/m3 Groundwater table 0.61 m 1.22 m sat 18.55 kN/m3 c 0 35 0.61 m 1.52 m Figure P3.11 450 kN 70 kN•m 16 kN/m3 c 0 30 1.2 m BB Water table sat 19 kN/m3 c 0 30 Figure P3.12 3.13 The shallow foundation shown in Figure 3.19 measures 1.2 m 3 1.8 m and is subjected to a centric load and a moment. If eB 5 0.12 m, eL 5 0.36 m, and the depth of the foundation is 1 m, determine the allowable load the foundation can carry. Use a factor of safety of 3. For the soil, we are told that unit weight g 5 17 kN>m3, friction angle fr 5 35°, and cohesion cr 5 0. References BOZOZUK, M. (1972). “Foundation Failure of the Vankleek Hill Tower Site,” Proceedings, Specialty Conference on Performance of Earth and Earth-Supported Structures, Vol. 1, Part 2, pp. 885–902. BRAND, E. W., MUKTABHANT, C., and TAECHANTHUMMARAK, A. (1972). “Load Test on Small Foundations in Soft Clay,” Proceedings, Specialty Conference on Performance of Earth and EarthSupported Structures, American Society of Civil Engineers, Vol. 1, Part 2, pp. 903–928. CAQUOT, A., and KERISEL, J. (1953). “Sur le terme de surface dans le calcul des fondations en milieu pulverulent,” Proceedings, Third International Conference on Soil Mechanics and Foundation Engineering, Zürich, Vol. I, pp. 336–337. DE BEER, E. E. (1970). “Experimental Determination of the Shape Factors and Bearing Capacity Factors of Sand,” Geotechnique, Vol. 20, No. 4, pp. 387– 411. HANNA, A. M., and MEYERHOF, G. G. (1981). “Experimental Evaluation of Bearing Capacity of Footings Subjected to Inclined Loads,” Canadian Geotechnical Journal, Vol. 18, No. 4, pp. 599–603. 180 Chapter 3: Shallow Foundations: Ultimate Bearing Capacity HANSEN, J. B. (1970). A Revised and Extended Formula for Bearing Capacity, Bulletin 28, Danish Geotechnical Institute, Copenhagen. HIGHTER, W. H., and ANDERS, J. C. (1985). “Dimensioning Footings Subjected to Eccentric Loads,” Journal of Geotechnical Engineering, American Society of Civil Engineers, Vol. 111, No. GT5, pp. 659–665. KUMBHOJKAR, A. S. (1993). “Numerical Evaluation of Terzaghi’s Ng,” Journal of Geotechnical Engineering, American Society of Civil Engineers, Vol. 119, No. 3, pp. 598–607. MEYERHOF, G. G. (1953). “The Bearing Capacity of Foundations Under Eccentric and Inclined Loads,” Proceedings, Third International Conference on Soil Mechanics and Foundation Engineering, Zürich, Vol. 1, pp. 440–445. MEYERHOF, G. G. (1963). “Some Recent Research on the Bearing Capacity of Foundations,” Canadian Geotechnical Journal, Vol. 1, No. 1, pp. 16–26. PRAKASH, S., and SARAN, S. (1971). “Bearing Capacity of Eccentrically Loaded Footings,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 97, No. SM1, pp. 95–117. PRANDTL, L. (1921). “Über die Eindringungsfestigkeit (Härte) plastischer Baustoffe und die Festigkeit von Schneiden,” Zeitschrift für angewandte Mathematik und Mechanik, Vol. 1, No. 1, pp. 15–20. PURKAYASTHA, R.D., and CHAR, R. A. N. (1977). “Stability Analysis of Eccentrically Loaded Footings,” Journal of Geotechnical Engineering Div., ASCE, Vol. 103, No. 6, pp. 647–651. REISSNER, H. (1924). “Zum Erddruckproblem,” Proceedings, First International Congress of Applied Mechanics, Delft, pp. 295–311. SARAN, S., and AGARWAL, R. B. (1991). “Bearing Capacity of Eccentrically Obliquely Loaded Footing,” Journal of Geotechnical Engineering, ASCE, Vol. 117, No. 11, pp. 1669–1690. TERZAGHI, K. (1943). Theoretical Soil Mechanics, Wiley, New York. VESIC, A. S. (1963). “Bearing Capacity of Deep Foundations in Sand,” Highway Research Record No. 39, National Academy of Sciences, pp. 112–153. VESIC, A. S. (1973). “Analysis of Ultimate Loads of Shallow Foundations,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 99, No. SM1, pp. 45–73. 4 4.1 Ultimate Bearing Capacity of Shallow Foundations: Special Cases Introduction The ultimate bearing capacity problems described in Chapter 3 assume that the soil supporting the foundation is homogeneous and extends to a great depth below the bottom of the foundation. They also assume that the ground surface is horizontal. However, that is not true in all cases: It is possible to encounter a rigid layer at a shallow depth, or the soil may be layered and have different shear strength parameters. In some instances, it may be necessary to construct foundations on or near a slope, or it may be required to design a foundation subjected to uplifting load. This chapter discusses bearing capacity problems relating to these special cases. 4.2 Foundation Supported by a Soil with a Rigid Base at Shallow Depth Figure 4.1(a) shows a shallow, rough continuous foundation supported by a soil that extends to a great depth. Neglecting the depth factor, for vertical loading Eq. (3.19) will take the form qu 5 crNc 1 qNq 1 1 gBNg 2 (4.1) The general approach for obtaining expressions for Nc, Nq, and Ng was outlined in Chapter 3. The extent of the failure zone in soil, D, at ultimate load obtained in the derivation of Nc and Nq by Prandtl (1921) and Reissner (1924) is given in Figure 4.1(b). Similarly, the magnitude of D obtained by Lundgren and Mortensen (1953) in evaluating Ng is given in the figure. Now, if a rigid, rough base is located at a depth of H , D below the bottom of the foundation, full development of the failure surface in soil will be restricted. In such a case, the soil failure zone and the development of slip lines at ultimate load will be as shown in Figure 4.2. 181 182 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases B qu q = Df Df 45 – /2 45 + /2 c D (a) 3 Nc and Nq D/B 2 N 1 0 0 10 20 30 40 Soil friction angle, (deg) 50 (b) Figure 4.1 (a) Failure surface under a rough continuous foundation; (b) variation of D>B with soil friction angle fr B qu q = Df H c Figure 4.2 Failure surface under a rough, continuous foundation with a rigid, rough base located at a shallow depth 4.2 Foundation Supported by a Soil with a Rigid Base at Shallow Depth 183 Mandel and Salencon (1972) determined the bearing capacity factors applicable to this case by numerical integration, using the theory of plasticity. According to their theory, the ultimate bearing capacity of a rough continuous foundation with a rigid, rough base located at a shallow depth can be given by the relation qu 5 crN *c 1 qN *q 1 1 gBN *g 2 (4.2) where N *c , N *q , N *g 5 modified bearing capacity factors B 5 width of foundation g 5 unit weight of soil Note that, for H > D, N*c 5 Nc , N*q 5 Nq , and N*g 5 Ng (Lundgren and Mortensen, 1953). The variations of N*c , N*q , and N*g with H>B and the soil friction angle fr are given in Figures 4.3, 4.4, and 4.5, respectively. 10,000 5000 H/B = 0.25 2000 0.33 1000 500 0.5 200 N *c 100 D/B = 2.4 1.0 50 1.6 20 1.2 10 5 0.9 0.7 2 1 0 10 20 (deg) 30 40 Figure 4.3 Mandel and Salencon’s bearing capacity factor N *c [Eq. (4.2)] 184 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases 10,000 5000 2000 1000 0.4 H/B = 0.2 500 0.6 200 N *q D/B = 3.0 1.0 100 2.4 50 1.9 20 1.6 10 5 1.4 1.2 2 1 20 25 30 35 (deg) 40 45 Figure 4.4 Mandel and Salencon’s bearing capacity factor N*q [Eq. (4.2)] Rectangular Foundation on Granular Soil Neglecting the depth factors, the ultimate bearing capacity of rough circular and rectangular foundations on a sand layer (cr 5 0) with a rough, rigid base located at a shallow depth can be given as qu 5 qN*q F*qs 1 1 gBN*g F*gs 2 (4.3) * * , Fgs 5 modified shape factors. where Fqs The shape factors F*qs and F*gs are functions of H>B and fr. On the basis of the work of Meyerhof and Chaplin (1953), and simplifying the assumption that, in radial planes, the stresses and shear zones are identical to those in transverse planes, Meyerhof (1974) proposed that 4.2 Foundation Supported by a Soil with a Rigid Base at Shallow Depth 185 10,000 5000 2000 1000 H/B = 0.2 500 0.4 200 0.6 N * D/B = 1.5 1.0 100 1.2 50 1.0 20 0.8 10 0.6 5 2 0.5 Figure 4.5 Mandel and Salencon’s bearing capacity factor Ng* [Eq. (4.2)] 1 20 25 30 35 (deg) 40 45 F*qs < 1 2 m1 ¢ B ≤ L (4.4) F*gs < 1 2 m2 ¢ B ≤ L (4.5) and where L 5 length of the foundation. The variations of m1 and m2 with H>B and fr are shown in Figure 4.6. More recently, Cerato and Lutenegger (2006) provided some test results for the bearing capacity factor, N*. These tests were conducted using square and circular plates with B varying from 0.152 m (6 in.) to 0.305 m (12 in.). It was assumed that Terzaghi’s bearingcapacity equations for square and circular foundations can be used. Or, from Eqs. (3.10) and (3.11) with c 0, qu qNq* 0.4BN* (square foundation) (4.6) 186 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases 1.0 H/B = 0.1 0.8 0.2 0.6 m1 0.4 0.4 0.6 1.0 0.2 2.0 0 20 25 35 30 (deg) 40 45 1.0 H/B = 0.1 0.8 0.2 0.4 0.6 m2 0.6 1.0 0.4 0.2 0 20 25 35 30 (deg) 40 45 Figure 4.6 Variation of m1 and m2 with H>B and fr and qu qNq* 0.3BN* (circular foundation) (4.7) The experimentally determined variation of N* is shown in Figure 4.7. It also was observed in this study that N* becomes equal to N at H/B ⬇ 3 instead of D/B, as shown in Figure 4.5. For that reason, Figure 4.7 shows the variation of N* for H/B 0.5 to 3.0. Foundation on Saturated Clay For saturated clay (i.e., under the undrained condition, or f 5 0), Eq. (4.2) will simplify to the form qu 5 cuN *c 1 q (4.8) 4.2 Foundation Supported by a Soil with a Rigid Base at Shallow Depth 187 2000 1500 N *γ 1000 H/B = 0.5 1.0 500 2.0 3.0 0 20 25 35 30 Friction angle, φ (deg) 40 45 Figure 4.7 Cerato and Lutenegger’s test results for N* Mandel and Salencon (1972) performed calculations to evaluate N *c for continuous foundations. Similarly, Buisman (1940) gave the following relationship for obtaining the ultimate bearing capacity of square foundations: qu(square) 5 ¢ p 1 2 1 B "2 2 ≤ cu 1 q 2H 2 ¢ for B "2 2 > 0≤ 2H 2 (4.9) In this equation, cu is the undrained shear strength. Equation (4.9) can be rewritten as B 2 0.707 H 5 5.14 £1 1 ≥ cu 1 q 5.14 (''''')''' ''* 0.5 qu(square) N*c(square) Table 4.1 gives the values of N *c for continuous and square foundations. (4.10) 188 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases Table 4.1 Values of N c* for Continuous and Square Foundations (f 5 0) N *c B H 2 3 4 5 6 8 10 a b Squarea Continuousb 5.43 5.93 6.44 6.94 7.43 8.43 9.43 5.24 5.71 6.22 6.68 7.20 8.17 9.05 Buisman’s analysis (1940) Mandel and Salencon’s analysis (1972) Example 4.1 A square foundation measuring 0.76 m 3 0.76 m is constructed on a layer of sand. We are given that Df 5 0.61 m, g 5 17.29 kN>m3, fr 5 35°, and cr 5 0. A rock layer is located at a depth of 0.46 m below the bottom of the foundation. Using a factor of safety of 4, determine the gross allowable load the foundation can carry. Solution From Eq. (4.3), qu 5 qN *qF *qs 1 1 gBN *gF *gs 2 and we also have q 5 17.29 3 0.61 5 10.55 kN>m2 For fr 5 35°, H>B 5 0.46>0.76 m 5 0.6, N *q < 90 (Figure 4.4), and N *g < 50 (Figure 4.5), and we have F *qs 5 1 2 m1 (B>L) From Figure 4.6(a), for fr 5 35°, H>B 5 0.6, and the value of m1 5 0.34, so F *qs 5 1 2 (0.34) (0.76>0.76) 5 0.66 Similarly, F *gs 5 1 2 m2 (B>L) From Figure 4.6(b), m2 5 0.45, so F *gs 5 1 2 (0.45) (0.76>0.76) 5 0.55 4.2 Foundation Supported by a Soil with a Rigid Base at Shallow Depth 189 Hence, qu 5 (10.55) (90) (0.66) 1 (1>2) (17.29) (0.76) (50) (0.55) 5 807.35 kN>m2 and Qall 5 quB2 (807.35) (0.76 3 0.76) 5 5 116.58 kN FS 4 ■ Example 4.2 Solve Example 4.1 using Eq. (4.6). Solution From Eq. (4.6), qu qNq* 0.4BN* For 35° and H/B 0.6, the value of Nq* ⬇ 90 (Figure 4.4) and N* ⬇ 230 (Figure 4.7). So qu (10.55)(90) (0.4)(17.29)(0.76)(230) 949.5 1208.9 2158.4 kN/m2 Q all 5 quB2 < 311.7 kN FS ■ Example 4.3 Consider a square foundation 1 m 3 1 m in plan located on a saturated clay layer underlain by a layer of rock. Given: Clay: cu 5 72 kN>m2 Unit weight: g 5 18 kN>m3 Distance between the bottom of foundation and the rock layer 5 0.25 m Df 5 1 m Estimate the gross allowable bearing capacity of the foundation. Use FS 3. Solution From Eq. (4.10), 0.5 qu 5 5.14 £1 1 B 2 0.707 H ≥ cu 1 q 5.14 190 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases For B>H 5 1>0.25 5 4; cu 5 72 kN>m2; and q 5 g Df 5 (18) (1) 5 18 kN>m3. qu 5 5.14B1 1 (0.5) (4) 2 0.707 R72 1 18 5 481.2 kN>m2 5.14 qall 5 4.3 qu 481.2 5 5 160.4 kN , m2 FS 3 ■ Bearing Capacity of Layered Soils: Stronger Soil Underlain by Weaker Soil The bearing capacity equations presented in Chapter 3 involve cases in which the soil supporting the foundation is homogeneous and extends to a considerable depth. The cohesion, angle of friction, and unit weight of soil were assumed to remain constant for the bearing capacity analysis. However, in practice, layered soil profiles are often encountered. In such instances, the failure surface at ultimate load may extend through two or more soil layers, and a determination of the ultimate bearing capacity in layered soils can be made in only a limited number of cases. This section features the procedure for estimating the bearing capacity for layered soils proposed by Meyerhof and Hanna (1978) and Meyerhof (1974). Figure 4.8 shows a shallow, continuous foundation supported by a stronger soil layer, underlain by a weaker soil that extends to a great depth. For the two soil layers, the physical parameters are as follows: Soil properties Layer Top Bottom Unit weight Friction angle Cohesion g1 g2 f1r f2r c1r c2r At ultimate load per unit area (qu ), the failure surface in soil will be as shown in the figure. If the depth H is relatively small compared with the foundation width B, a punching shear failure will occur in the top soil layer, followed by a general shear failure in the bottom soil layer. This is shown in Figure 4.8(a). However, if the depth H is relatively large, then the failure surface will be completely located in the top soil layer, which is the upper limit for the ultimate bearing capacity. This is shown in Figure 4.8b. The ultimate bearing capacity for this problem, as shown in Figure 4.8a, can be given as qu 5 qb 1 2(Ca 1 Pp sin dr) B 2 g1H (4.11) 4.3 Bearing Capacity of Layered Soils: Stronger Soil Underlain by Weaker Soil 191 B qu Df a b Ca H Ca Pp Pp a Stronger soil 1 1 c1 b (a) B Weaker soil 2 2 c2 qu Df Stronger soil 1 1 c1 H (b) Weaker soil 2 2 c2 Figure 4.8 Bearing capacity of a continuous foundation on layered soil where B 5 width of the foundation Ca 5 adhesive force Pp 5 passive force per unit length of the faces aar and bbr qb 5 bearing capacity of the bottom soil layer dr 5 inclination of the passive force Pp with the horizontal Note that, in Eq. (4.11), Ca 5 car H where car 5 adhesion. Equation (4.11) can be simplified to the form qu 5 qb 1 2Df KpH tan dr 2car H 1 g1H2 ¢1 1 ≤ 2 g1H B H B (4.12) where KpH 5 horizontal component of passive earth pressure coefficient. However, let KpH tan dr 5 Ks tan f1r (4.13) 192 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases where Ks 5 punching shear coefficient. Then, qu 5 qb 1 2Df Ks tan f1r 2car H 1 g1H2 ¢ 1 1 ≤ 2 g1H B H B (4.14) The punching shear coefficient, Ks , is a function of q2>q1 and fr1 , or, specifically, Ks 5 f¢ q2 , fr ≤ q1 1 Note that q1 and q2 are the ultimate bearing capacities of a continuous foundation of width B under vertical load on the surfaces of homogeneous thick beds of upper and lower soil, or q1 5 c1r Nc(1) 1 12g1BNg(1) (4.15) q2 5 c2r Nc(2) 1 12g2BNg(2) (4.16) and where Nc(1) , Ng(1) 5 bearing capacity factors for friction angle f1r (Table 3.3) Nc(2) , Ng(2) 5 bearing capacity factors for friction angle f2r (Table 3.3) Observe that, for the top layer to be a stronger soil, q2 >q1 should be less than unity. The variation of Ks with q2>q1 and f1r is shown in Figure 4.9. The variation of car >c1r with q2>q1 is shown in Figure 4.10. If the height H is relatively large, then the 40 30 KS q2 q1 = 1 20 0.4 10 0.2 0 0 20 30 40 1 (deg) 50 Figure 4.9 Meyerhof and Hanna’s punching shear coefficient Ks 4.3 Bearing Capacity of Layered Soils: Stronger Soil Underlain by Weaker Soil 193 1.0 0.9 ca 0.8 c1 0.7 0.6 0 0.2 0.4 0.6 0.8 1.0 q2 q1 Figure 4.10 Variation of car >c1r with q2>q1 based on the theory of Meyerhof and Hanna (1978) failure surface in soil will be completely located in the stronger upper-soil layer (Figure 4.8b). For this case, qu 5 qt 5 c1r Nc(1) 1 qNq(1) 1 12 g1BNg(1). (4.17) where Nc(1), Nq(1), and Ng(1) 5 bearing capacity factors for fr 5 f1r (Table 3.3) and q 5 g1Df. Combining Eqs. (4.14) and (4.17) yields qu 5 qb 1 2Df Ks tan f1r 2car H 1 g1H2 ¢1 1 ≤ 2 g1H < qt B H B (4.18) For rectangular foundations, the preceding equation can be extended to the form qu 5 qb 1 ¢1 1 B L ≤¢ 1 g1H2 ¢1 1 2car H B B L ≤ ≤ ¢1 1 2Df H ≤¢ Ks tan f1r B (4.19) ≤ 2 g1H < qt where qb 5 c2r Nc(2)Fcs(2) 1 g1 (Df 1 H)Nq(2)Fqs(2) 1 1 g BNg(2)Fgs(2) 2 2 (4.20) 194 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases and qt 5 c1r Nc(1)Fcs(1) 1 g1DfNq(1)Fqs(1) 1 1 g BNg(1)Fgs(1) 2 1 (4.21) in which Fcs(1) , Fqs(1) , Fgs(1) 5 shape factors with respect to top soil layer (Table 3.4) Fcs(2) , Fqs(2) , Fgs(2) 5 shape factors with respect to bottom soil layer (Table 3.4) Special Cases 1. Top layer is strong sand and bottom layer is saturated soft clay (f2 5 0). From Eqs. (4.19), (4.20), and (4.21), qb 5 ¢1 1 0.2 B ≤ 5.14c2 1 g1 (Df 1 H) L (4.22) and qt 5 g1DfNq(1)Fqs(1) 1 12g1BNg(1)Fgs(1) (4.23) 2Df Ks tan f1r B B ≤ 5.14c2 1 g1H2 ¢1 1 ≤ ¢ 1 1 ≤ L L H B 1 1 g1Df < g1DfNq(1)Fqs(1) 1 g1BNg(1)Fgs(1) 2 (4.24) Hence, qu 5 ¢ 1 1 0.2 where c2 5 undrained cohesion. For a determination of Ks from Figure 4.9, c2Nc(2) q2 5.14c2 51 5 q1 0.5g1BNg(1) 2 g1BNg(1) (4.25) 2. Top layer is stronger sand and bottom layer is weaker sand (cr1 5 0, cr2 5 0). The ultimate bearing capacity can be given as qu 5 Bg1 (Df 1 H)Nq(2)Fqs(2) 1 1 2 g2BNg(2)Fgs(2) R 2Df Ks tan f1r 1 g1H2 ¢1 1 ≤ ¢1 1 ≤ 2 g1H < qt L H B B (4.26) 4.3 Bearing Capacity of Layered Soils: Stronger Soil Underlain by Weaker Soil 195 where qt 5 g1DfNq(1)Fqs(1) 1 1 g BN F 2 1 g(1) gs(1) (4.27) Then 1 g2Ng(2) q2 2 g2BNg(2) 51 5 q1 g1Ng(1) 2 g1BNg(1) (4.28) 3. Top layer is stronger saturated clay (f1 5 0) and bottom layer is weaker saturated clay (f2 5 0). The ultimate bearing capacity can be given as qu 5 ¢1 1 0.2 B B 2caH ≤5.14c2 1 ¢ 1 1 ≤ ¢ ≤ 1 g1Df < qt L L B (4.29) B ≤5.14c1 1 g1Df L (4.30) where qt 5 ¢1 1 0.2 and c1 and c2 are undrained cohesions. For this case, 5.14c2 c2 q2 5 5 q1 c1 5.14c1 (4.31) Example 4.4 Refer to Figure 4.8(a) and consider the case of a continuous foundation with B 5 2 m, Df 5 1.2 m , and H 5 1.5 m. The following are given for the two soil layers: Top sand layer: Unit weight g1 5 17.5 kN>m3 f1r 5 40° c1r 5 0 Bottom clay layer: Unit weight g2 5 16.5 kN>m3 fr2 5 0 c2 5 30 kN>m2 Determine the gross ultimate load per unit length of the foundation. 196 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases Solution For this case, Eqs. (4.24) and (4.25) apply. For fr1 5 40°, from Table 3.3, Ng 5 109.41 and c2Nc(2) q2 (30) (5.14) 5 5 5 0.081 q1 0.5g1BNg(1) (0.5)(17.5)(2)(109.41) From Figure 4.9, for c2Nc(2)>0.5g1BNg(1) 5 0.081 and f1r 5 40°, the value of Ks < 2.5 . Equation (4.24) then gives qu 5 B1 1 (0.2) ¢ 2Df tan f1r B B ≤ R 5.14c2 1 ¢ 1 1 ≤g1H 2 ¢1 1 ≤ Ks 1 g1Df L L H B 5 31 1 (0.2) (0) 4 (5.14) (30) 1 (1 1 0) (17.5) (1.5) 2 3 B1 1 (2) (1.2) tan 40 R (2.5) 1 (17.5) (1.2) 1.5 2.0 5 154.2 1 107.4 1 21 5 282.6 kN>m2 Again, from Eq. (4.27), qt 5 g1DfNq(1)Fqs(1) 1 12 g1BNg(1)Fgs(1) From Table 3.3, for fr1 5 40°, Ng 5 109.4 and Nq 5 64.20 . From Table 3.4, Fqs(1) 5 1 1 ¢ B ≤ tan f1r 5 1 1 (0)tan 40 5 1 L and Fgs(1) 5 1 2 0.4 B 5 1 2 (0.4) (0) 5 1 L so that qt 5 (17.5) (1.2) (64.20) (1) 1 ( 12 ) (17.5) (2) (109.4) (1) 5 3262.7 kN>m2 Hence, qu 5 282.6 kN>m2 Qu 5 (282.6) (B) 5 (282.6) (2) 5 565.2 kN , m ■ 4.3 Bearing Capacity of Layered Soils: Stronger Soil Underlain by Weaker Soil 197 Example 4.5 A foundation 1.5 m 3 1 m is located at a depth Df of 1 m in a stronger clay. A softer clay layer is located at a depth H of 3 ft, measured from the bottom of the foundation. For the top clay layer, Undrained shear strength 5 120 kN>m2 Unit weight 5 16.8 kN>m3 and for the bottom clay layer, Undrained shear strength 5 48 kN>m2 Unit weight 5 16.2 kN>m3 Determine the gross allowable load for the foundation with an FS of 3. Solution For this problem, Eqs. (4.29), (4.30), and (4.31) will apply, or qu 5 ¢1 1 0.2 B B 2caH ≤5.14c2 1 ¢ 1 1 ≤ ¢ ≤ 1 g1Df L L B < ¢1 1 0.2 B ≤ 5.14c1 1 g1Df L We are given the following data: B51m H51m L 5 1.5 m Df 5 1 m g1 5 16.8 kN>m3 From Figure 4.10 for c2>c1 5 48>120 5 0.4, the value of ca>c1 < 0.9, so ca 5 (0.9) (2500) 5 108 kN>m2 and qu 5 B1 1 (0.2) ¢ (2) (108) (1) 1 1 ≤ R (5.14) (48) 1 ¢ 1 1 ≤B R 1 (16.8) (1) 1.5 1.5 1 5 279.6 1 360 1 16.8 5 656.4 kN>m2 As a check, we have, from Eq. (4.30), qt 5 B1 1 (0.2) ¢ 1 ≤ R (5.14) (120) 1 (16.8) (1) 1.5 5 699 1 16.8 5 715.8 kN>m2 Thus, qu 5 656.4 kN>m2 (i.e., the smaller of the two values just calculated), and qall 5 qu 656.4 5 5 218.8 kN>m2 FS 3 The total allowable load is therefore (qall ) (1 3 1.5) 5 328.2 kN ■ 198 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases 4.4 Bearing Capacity of Layered Soil: Weaker Soil Underlain by Stronger Soil When a foundation is supported by a weaker soil layer underlain by a stronger layer (Figure 4.11a), the ratio of q2/q1 defined by Eqs. (4.15) and (4.16) will be greater than one. Also, if H/B is relatively small, as shown in the left-hand half of Figure 4.11a, the failure surface in soil at ultimate load will pass through both soil layers. However, for larger H/B ratios, the failure surface will be fully located in the top, weaker soil layer, as shown in the Weaker soil γ1 φ ′1 Df c′1 B H D H Stronger soil γ2 φ ′2 c′2 Stronger soil γ2 φ ′2 (a) c′2 qu qb qt D/B (b) H/B Figure 4.11 (a) Foundation on weaker soil layer underlain by stronger sand layer, (b) Nature of variation of qu with H/B 199 4.4 Bearing Capacity of Layered Soil: Weaker Soil Underlain by Stronger Soil right-hand half of Figure 4.11a. For this condition, the ultimate bearing capacity (Meyerhof, 1974; Meyerhof and Hanna, 1978) can be given by the empirical equation qu 5 qt 1 (qb 2 qt ) a H 2 b $ qt D (4.32) where D depth of failure surface beneath the foundation in the thick bed of the upper weaker soil layer qt ultimate bearing capacity in a thick bed of the upper soil layer qb ultimate bearing capacity in a thick bed of the lower soil layer So 1 qt 5 c1Nc(1)Fcs(1) 1 g1DfNq(1)Fqs(1) 1 g1BNg(1)Fgs(1) 2 (4.33) 1 qt 5 c2Nc(2)Fcs(2) 1 g2DfNq(2)Fqs(2) 1 g2BNg(2)Fgs(2) 2 (4.34) and where Nc(1), Nq(1), N(1) bearing capacity factors corresponding to the soil friction angle 1 Nc(2), Nq(2), N(2) bearing capacity factors corresponding to the soil friction angle 2 Fcs(1), Fqs(1), Fs(1) shape factors corresponding to the soil friction angle 1 Fcs(2), Fqs(2), Fs(2) shape factors corresponding to the soil friction angle 2 Meyerhof and Hanna (1978) suggested that • • D ⬇ B for loose sand and clay D ⬇ 2B for dense sand Equations (4.32), (4.33), and (4.34) imply that the maximum and minimum values of qu will be qb and qt, respectively, as shown in Figure 4.11b. Example 4.6 Refer to Figure 4.11a. For a layered saturated-clay profile, given: L 1.83 m, B 1.22 m, Df 0.91 m, H 0.61 m, 1 17.29 kN/m3, 1 0, c1 57.5 kN/m2, 2 19.65 kN/m3, 2 0, and c2 119.79 kN/m2. Determine the ultimate bearing capacity of the foundation. Solution From Eqs. (4.15) and (4.16), q2 c2Nc c2 119.79 5 5 5 5 2.08 . 1 q1 c1 c1Nc 57.5 So, Eq. (4.32) will apply. From Eqs. (4.33) and (4.34) with 1 2 0, B qt 5 a1 1 0.2 bNcc1 1 g1Df L 5 c1 1 (0.2) a 1.22 b d (5.14) (57.5) 1 (0.91) (17.29) 5 334.96 1 15.73 1.83 5 350.69 kN>m2 200 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases and qb 5 a1 1 0.2 B bN c 1 g2Df L c 2 5 c1 1 (0.2) a 1.22 b d (5.14) (119.79) 1 (0.91) (19.65) 1.83 5 697.82 1 17.88 5 715.7 kN>m2 From Eq. (4.32), qu 5 qt 1 (qb 2 qt ) a H 2 b D D<B qu 5 350.69 1 (715.7 2 350.69) a 0.61 2 b < 442 kN>m2 . qt 1.22 Hence, qu 442 kN/m2 4.5 ■ Closely Spaced Foundations—Effect on Ultimate Bearing Capacity In Chapter 3, theories relating to the ultimate bearing capacity of single rough continuous foundations supported by a homogeneous soil extending to a great depth were discussed. However, if foundations are placed close to each other with similar soil conditions, the ultimate bearing capacity of each foundation may change due to the interference effect of the failure surface in the soil. This was theoretically investigated by Stuart (1962) for granular soils. It was assumed that the geometry of the rupture surface in the soil mass would be the same as that assumed by Terzaghi (Figure 3.5). According to Stuart, the following conditions may arise (Figure 4.12). Case I. (Figure 4.12a) If the center-to-center spacing of the two foundations is x $ x1, the rupture surface in the soil under each foundation will not overlap. So the ultimate bearing capacity of each continuous foundation can be given by Terzaghi’s equation [Eq. (3.3)]. For (cr 5 0) qu 5 qNq 1 12 gBNg (4.35) where Nq, Ng 5 Terzaghi’s bearing capacity factors (Table 3.1). Case II. (Figure 4.12b) If the center-to-center spacing of the two foundations (x 5 x2 , x1 ) are such that the Rankine passive zones just overlap, then the magnitude of qu will still be given by Eq. (4.35). However, the foundation settlement at ultimate load will change (compared to the case of an isolated foundation). Case III. (Figure 4.12c) This is the case where the center-to-center spacing of the two continuous foundations is x 5 x3 , x2. Note that the triangular wedges in the soil 4.5 Closely Spaced Foundations—Effect on Ultimate Bearing Capacity 201 x x1 B qu 2 2 B qu 2 2 1 2 2 q Df 2 2 1 (a) x x2 B qu B qu 2 2 2 2 1 1 q Df 2 2 (b) x x3 2 g1 B qu B qu 3 3 e d1 d2 (c) q Df 2 g2 x x4 B qu B qu q Df (d) Figure 4.12 Assumptions for the failure surface in granular soil under two closely spaced rough continuous foundations (Note: a1 5 fr, a2 5 45 2 fr> 2, a3 5 180 2 2fr) under the foundations make angles of 180° 2 2fr at points d1 and d2. The arcs of the logarithmic spirals d1g1 and d1e are tangent to each other at d1. Similarly, the arcs of the logarithmic spirals d2g2 and d2e are tangent to each other at d2. For this case, the ultimate bearing capacity of each foundation can be given as (cr 5 0) qu 5 qNqzq 1 21 gBNg zg where zq, zg 5 efficiency ratios. (4.36) 202 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases 2.0 Rough base Along this line two footings act as one = 40 q 1.5 37 39 35 32 30 1.0 1 2 4 3 x/B (a) 5 3.5 Rough base Along this line two footings act as one 3.0 2.5 = 40 39 2.0 37 35 32 1.5 30 1.0 1 2 3 x/B (b) 4 5 Figure 4.13 Variation of efficiency ratios with x>B and fr The efficiency ratios are functions of x>B and soil friction angle fr. The theoretical variations of zq and zg are given in Figure 4.13. Case IV. (Figure 4.12d): If the spacing of the foundation is further reduced such that x 5 x4 , x3, blocking will occur and the pair of foundations will act as a single foundation. The soil between the individual units will form an inverted arch which travels down with the foundation as the load is applied. When the two foundations touch, the zone of 203 4.6 Bearing Capacity of Foundations on Top of a Slope arching disappears and the system behaves as a single foundation with a width equal to 2B. The ultimate bearing capacity for this case can be given by Eq. (4.35), with B being replaced by 2B in the second term. The ultimate bearing capacity of two continuous foundations spaced close to each other may increase since the efficiency ratios are greater than one. However, when the closely spaced foundations are subjected to a similar load per unit area, the settlement Se will be larger when compared to that for an isolated foundation. 4.6 Bearing Capacity of Foundations on Top of a Slope In some instances, shallow foundations need to be constructed on top of a slope. In Figure 4.14, the height of the slope is H, and the slope makes an angle b with the horizontal. The edge of the foundation is located at a distance b from the top of the slope. At ultimate load, qu , the failure surface will be as shown in the figure. Meyerhof (1957) developed the following theoretical relation for the ultimate bearing capacity for continuous foundations: qu 5 crNcq 1 12 gBNgq (4.37) For purely granular soil, cr 5 0, thus, qu 5 12 gBNgq (4.38) Again, for purely cohesive soil, f 5 0 (the undrained condition); hence, qu 5 cNcq (4.39) where c 5 undrained cohesion. b B qu Df H c Figure 4.14 Shallow foundation on top of a slope 204 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases 400 Df B 300 Df =0 B =1 0 200 Nq 20 = 40 40 = 40 100 0 20 0 50 = 30 30 40 25 = 30 0 10 5 30 1 0 1 2 3 b B 4 5 6 Figure 4.15 Meyerhof’s bearing capacity factor Ngq for granular soil (cr 5 0) The variations of Ngq and Ncq defined by Eqs. (4.38) and (4.39) are shown in Figures 4.15 and 4.16, respectively. In using Ncq in Eq. (4.39) as given in Figure 4.16, the following points need to be kept in mind: 1. The term Ns 5 gH c (4.40) is defined as the stability number. 2. If B , H, use the curves for Ns 5 0. 3. If B > H, use the curves for the calculated stability number Ns . Stress Characteristics Solution for Granular Soil Slopes For slopes in granular soils, the ultimate bearing capacity of a continuous foundation can be given by Eq. (4.38), or 1 qu 5 gBNgq 2 On the basis of the method of stress characteristics, Graham, Andrews, and Shields (1988) provided a solution for the bearing capacity factor N q for a shallow continuous foundation on the top of a slope in granular soil. Figure 4.17 shows the schematics of the failure zone in the soil for embedment (Df/B) and setback (b/B) assumed for those authors’ analysis. The variations of N q obtained by this method are shown in Figures 4.18, 4.19, and 4.20. 4.6 Bearing Capacity of Foundations on Top of a Slope 8 6 Df 0 B Ns 0 Df 1 B 0 15 30 45 60 90 Ns 0 0 30 60 Ncq 205 90 4 Ns 2 0 30 60 90 2 Ns 4 0 30 60 90 0 2 3 4 1 b for N 0; b for N 0 s s H B 0 5 Figure 4.16 Meyerhof’s bearing capacity factor Ncq for purely cohesive soil B Df (a) b (b) Figure 4.17 Schematic diagram of failure zones for embedment and setback: (a) Df /B 0; (b) b/B 0 206 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases 1000 1000 b/B = 0 b/B = 0.5 b/B = 1 b/B = 2 = 45° 40° 40° Nq Nq = 45° 100 100 35° 35° 30° 30° 10 0 10 20 30 10 40 0 10 20 (deg) (deg) (a) (b) 30 40 Figure 4.18 Graham et al.’s theoretical values of Nq(Df/B 0) 1000 1000 b/B = 0 b/B = 0.5 = 45° 40° Nq Nq 40° 100 35° 35° 100 30° 30° 10 0 b/B = 1 b/B = 2 = 45° 10 20 30 40 10 0 10 20 (deg) (deg) (a) (b) Figure 4.19 Graham et al.’s theoretical values of Nq(Df/B 0.5) 30 40 4.6 Bearing Capacity of Foundations on Top of a Slope 207 1000 1000 = 45° b/B = 0 b/B = 0.5 = 45° b/B = 1 b/B = 2 40° 40° Nq Nq 35° 35° 100 100 30° 30° 10 0 10 20 30 10 40 0 10 20 (deg) (deg) (a) (b) 30 40 Figure 4.20 Graham et al.’s theoretical values of Nq(Df/B 1) Example 4.7 In Figure 4.14, for a shallow continuous foundation in a clay, the following data are given: B 5 1.2 m; Df 5 1.2 m; b 5 0.8 m; H 5 6.2 m; b 5 30°; unit weight of soil 5 17.5 kN>m3 ; f 5 0; and c 5 50 kN>m2 . Determine the gross allowable bearing capacity with a factor of safety FS 5 4. Solution Since B , H, we will assume the stability number Ns 5 0. From Eq. (4.39), qu 5 cNcq We are given that Df B 5 1.2 51 1.2 and b 0.8 5 5 0.67 B 1.2 208 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases For b 5 30°, Df>B 5 1 and b>B 5 0.67, Figure 4.16 gives Ncq 5 6.3. Hence, qu 5 (50) (6.3) 5 315 kN>m2 and qall 5 qu 315 5 5 78.8 kN , m2 FS 4 ■ Example 4.8 Figure 4.21 shows a continuous foundation on a slope of a granular soil. Estimate the ultimate bearing capacity. 1.5 m 1.5 m 1.5 m 6m 30º 16.8 kN/m3 40º c 0 Figure 4.21 Foundation on a granular slope Solution For granular soil (cr 5 0), from Eq. (4.38), qu 5 12 gBNgq We are given that b>B 5 1.5>1.5 5 1, Df>B 5 1.5>1.5 5 1, fr 5 40°, and b 5 30°. From Figure 4.15, Ngq < 120. So, qu 5 12 (16.8) (1.5) (120) 5 1512 kN , m2 ■ Example 4.9 Solve Example 4.8 using the stress characteristics solution method. Solution 1 qu 5 gBNgq 2 From Figure 4.20b, Nq ⬇ 110. Hence, 1 qu 5 (16.8) (1.5) (110) 5 1386 kN>m2 2 ■ 4.7 Seismic Bearing Capacity of a Foundation at the Edge of a Granular Soil Slope 4.7 209 Seismic Bearing Capacity of a Foundation at the Edge of a Granular Soil Slope Figure 4.22 shows a continuous surface foundation (B/L 0, Df/B 0) at the edge of a granular slope. The foundation is subjected to a loading inclined at an angle to the vertical. Let the foundation be subjected to seismic loading with a horizontal coefficient of acceleration, kh. Based on their analysis of method of slices, Huang and Kang (2008) expressed the ultimate bearing capacity as 1 qu 5 gBNgFgiFgbFge 2 (4.41) where N bearing capacity factor (Table 3.3) Fi load inclination factor F slope inclination factor Fe correction factor for the inertia force induced by seismic loading The relationships for Fi, F , and Fe are as follow: Fgi 5 c1 2 a a° (0.1f921.21) bd fr° (4.42) Fgb 5 31 2 (1.062 2 0.014f9 )tanf r4 a 10° b b° (4.43) and Fe 1 [(2.57 0.043)e1.45 tan ]kh (4.44) In Eqs. (4.42) through (4.44), is in degrees. qu B Sand c = 0 Figure 4.22 Continuous foundation at the edge of a granular slope subjected to seismic loading 210 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases Example 4.10 Consider a continuous surface foundation on a granular soil slope subjected to a seismic loading, as shown in Figure 4.22. Given: B 1.5 m, 17.5 kN/m3, 35°, c 0, 30°, 10°, and kh 0.2. Calculate the ultimate bearing capacity, qu. Solution From Eq. (4.41), 1 qu 5 gBNgFgiFgbFge 2 For 35°, N 48.03 (Table 3.3). Thus, Fgt 5 c1 2 a a° (0.1f921.21) 10 3(0.1335)21.214 b d 5 c1 2 a bd 5 0.463 35 f9° Fgb 5 31 2 (1.062 2 0.014f9 )tanf r4 a 10° b b° 5 31 2 (1.062 2 0.014 3 35)tan 354 a 10 b 5 0.215 30 Fe 1 3(2.57 3(2.57 1 0.043)e1.45 tan 4 kh 0.043 35)e1.45 tan 304 (0.2) 0.508 So 1 qu 5 (17.5) (1.5) (48.03) (0.463) (0.215) (0.508) 5 31.9 kN>m2 2 4.8 ■ Bearing Capacity of Foundations on a Slope A theoretical solution for the ultimate bearing capacity of a shallow foundation located on the face of a slope was developed by Meyerhof (1957). Figure 4.23 shows the nature of the plastic zone developed under a rough continuous foundation of width B. In Figure 4.23, abc is an elastic zone, acd is a radial shear zone, and ade is a mixed shear zone. Based on this solution, the ultimate bearing capacity can be expressed as B qu Df a b e 90 d 90 c Figure 4.23 Nature of plastic zone under a rough continuous foundation on the face of a slope 211 4.8 Bearing Capacity of Foundations on a Slope qu 5 cNcqs (for purely cohesive soil, that is, f 5 0) (4.45) qu 5 12 gBNgqs (for granular soil, that is cr 5 0) (4.46) and The variations of Ncqs and Ngqs with slope angle b are given in Figures 4.24 and 4.25. 8 Df /B 0 Df /B 1 7 6 Ns 0 Ncqs 5 4 0 3 1 2 2 3 4 1 5 5.53 0 0 20 40 60 (deg) Figure 4.24 Variation of Ncqs with b. (Note: Ns 5 gH>c) 80 600 500 Df /B 0 Df /B 1 400 300 45 Nqs 200 40 100 45 50 30 25 40 10 5 1 0 30 0 10 20 30 (deg) 40 50 Figure 4.25 Variation of Ngqs with b 212 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases 4.9 Foundations on Rock On some occasions, shallow foundations may have to be built on rocks, as shown in Figure 4.26. For estimation of the ultimate bearing capacity of shallow foundations on rock, we may use Terzaghi’s bearing capacity equations [Eqs. (3.3), (3.7) and (3.8)] with the bearing capacity factors given here (Stagg and Zienkiewicz, 1968; Bowles, 1996): f9 b 2 Nc 5 5 tan4 a45 1 Nq 5 tan6 a45 1 f9 b 2 N Nq 1 (4.47) (4.48) (4.49) For rocks, the magnitude of the cohesion intercept, c, can be expressed as quc 5 2c9tana45 1 f9 b 2 (4.50) where quc unconfined compression strength of rock angle of friction The unconfined compression strength and the friction angle of rocks can vary widely. Table 4.2 gives a general range of quc for various types of rocks. It is important to keep in mind that the magnitude of quc and (hence c) reported from laboratory tests are for intact rock specimens. It does not account for the effect of discontinuities. To account for discontinuities, Bowles (1996) suggested that the ultimate bearing capacity qu should be modified as qu(modified) qu(RQD)2 where RQD rock quality designation (see Chapter 2). γ Soil Df B Rock c Figure 4.26 Foundation on rock Table 4.2 Range of the Unconfined Compression Strength of Various Types of Rocks quc Rock type MN/m2 kip/m2 (deg) Granite Limestone Sandstone Shale 65–250 30–150 25–130 5–40 9.5–36 4–22 3.5–19 0.75–6 45–55 35–45 30–45 15–30 (4.51) 4.10 Uplift Capacity of Foundations 213 In any case, the upper limit of the allowable bearing capacity should not exceed fc (28-day compressive strength of concrete). Example 4.11 Refer to Figure 4.26. A square column foundation is to be constructed over siltstone. Given: B 2.5 m 2.5 m Df 2 m 17 kN/m3 c 32 MN/m2 31° 25 kN/m3 RDQ 50% Foundation: B Soil: Siltstone: Estimate the allowable load-bearing capacity. Use FS 4. Also, for concrete, use fc 30 MN/m2. Solution From Eq. (3.7), qu 1.3cNc qNq 0.4BN Nc 5 5 tan4 a45 1 Nq 5 tan6 a45 1 f9 31 b 5 5 tan4 a45 1 b 5 48.8 2 2 f9 31 b 5 tan6 a45 1 b 5 30.5 2 2 N Nq 1 30.5 1 31.5 Hence, qu (1.3)(32 2030.08 2031.9 103 kN/m2)(48.8) (17 103 1.037 2)(30.5) (0.4)(25)(2.5)(31.5) 103 0.788 103 103 kN/m2 ⬇ 2032 MN/m2 qu(modified) qu(RQD)2 (2032)(0.5)2 508 MN/m2 q all 5 508 5 127 MN>m2 4 Since 127 MN/m2 is greater than fc, use qall 30 MN/m2. 4.10 ■ Uplift Capacity of Foundations Foundations may be subjected to uplift forces under special circumstances. During the design process for those foundations, it is desirable to provide a sufficient factor of safety against failure by uplift. This section will provide the relationships for the uplift capacity of foundations in granular and cohesive soils. 214 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases Foundations in Granular Soil (c9 5 0) Figure 4.27 shows a shallow continuous foundation that is being subjected to an uplift force. At ultimate load, Qu , the failure surface in soil will be as shown in the figure. The ultimate load can be expressed in the form of a nondimensional breakout factor, Fq. Or Fq 5 Qu AgDf (4.52) where A 5 area of the foundation. The breakout factor is a function of the soil friction angle fr and Df>B. For a given soil friction angle, Fq increases with Df>B to a maximum at Df>B 5 (Df>B) cr and remains constant thereafter. For foundations subjected to uplift, Df>B # (Df>B) cr is considered a shallow foundation condition. When a foundation has an embedment ratio of Df>B . (Df>B) cr , it is referred to as a deep foundation. Meyerhof and Adams (1968) provided relationships to estimate the ultimate uplifting load Qu for shallow 3that is, Df>B # (Df>B) cr4 , circular, and rectangular foundations. Using these relationships and Eq. (4.52), Das and Seeley (1975) expressed the breakout factor Fq in the following form Fq 5 1 1 2B1 1 m¢ Df B ≤Ra Df B bKu tan fr (4.53) (for shallow circular and square foundations) Fq 5 1 1 b B 1 1 2m¢ Df B ≤R¢ Df B ≤ 1 1r ¢ ≤Ku tan fr L B (for shallow rectangular foundations) Qu Pp Pp Sand Unit weight γ Friction angle Df B Figure 4.27 Shallow continuous foundation subjected to uplift (4.54) 4.10 Uplift Capacity of Foundations 215 where m 5 a coefficient which is a function of fr Ku 5 nominal uplift coefficient The variations of Ku, m, and (Df>B) cr for square and circular foundations are given in Table 4.3 (Meyerhof and Adams, 1968). For rectangular foundations, Das and Jones (1982) recommended that ¢ Df B ≤ 5 ¢ cr-rectangular Df B ≤ B0.133¢ cr-square Df L ≤ 1 0.867R # 1.4¢ ≤ B B cr-square (4.55) Using the values of Ku, m, and (Df>B) cr in Eq. (4.53), the variations of Fq for square and circular foundations have been calculated and are shown in Figure 4.28. A step-bystep procedure to estimate the uplift capacity of foundations in granular soil follows. Determine, Df, B, L, and fr. CalculateDf>B. Using Table 4.3 and Eq. (4.55), calculate (Df >B) cr. If Df>B is less than or equal to (Df>B) cr, it is a shallow foundation. If Df>B . (Df>B) cr, it is a deep foundation. For shallow foundations, use Df>B calculated in Step 2 in Eq. (4.53) or (4.54) to estimate Fq. Thus, Qu 5 Fq AgDf. Step 7. For deep foundations, substitute (Df>B) cr for Df>B in Eq. (4.53) or (4.54) to obtain Fq, from which the ultimate load Qu may be obtained. Step 1. Step 2. Step 3. Step 4. Step 5. Step 6. 100 45 40 Fq 35 10 30 1 1 2 3 4 5 6 Df /B 7 8 9 10 Figure 4.28 Variation of Fq with Df>B and fr 216 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases Table 4.3 Variation of Ku, m, and (Df>B) cr Soil friction angle, f9 (deg) Ku m (Df >B)cr for square and circular foundations 20 25 30 35 40 45 0.856 0.888 0.920 0.936 0.960 0.960 0.05 0.10 0.15 0.25 0.35 0.50 2.5 3 4 5 7 9 Foundations in Cohesive Soil (f9 5 0) The ultimate uplift capacity, Qu, of a foundation in a purely cohesive soil can be expressed as Qu 5 A(gDf 1 cuFc ) (4.56) where A 5 area of the foundation cu 5 undrained shear strength of soil Fc 5 breakout factor As in the case of foundations in granular soil, the breakout factor Fc increases with embedment ratio and reaches a maximum value of Fc 5 F c* at Df>B 5 (Df>B) cr and remains constant thereafter. Das (1978) also reported some model test results with square and rectangular foundations. Based on these test results, it was proposed that ¢ Df B ≤ 5 0.107cu 1 2.5 # 7 (4.57) cr-square where Df critical embedment ratio of square (or circular) foundations ¢ ≤ B cr-square cu 5 undrained cohesion, in kN>m2 It was also observed by Das (1980) that ¢ Df B ≤ 5 ¢ cr-rectangular Df B ≤ B0.73 1 0.27¢ cr-square Df L (4.58) ≤ R # 1.55¢ ≤ B B cr-square where Df ¢ ≤ critical embedment ratio of rectangular foundations B cr-rectangular L 5 length of foundation 217 4.10 Uplift Capacity of Foundations Based on these findings, Das (1980) proposed an empirical procedure to obtain the breakout factors for shallow and deep foundations. According to this procedure, ar and br are two nondimensional factors defined as Df ar 5 ¢ B Df B (4.59) ≤ cr and br 5 Fc (4.60) F c* For a given foundation, the critical embedment ratio can be calculated using Eqs. (4.57) and (4.58). The magnitude of F c* can be given by the following empirical relationship * F c-rectangular 5 7.56 1 1.44¢ B ≤ L (4.61) * 5 breakout factor for deep rectangular foundations where F c-rectangular Figure 4.29 shows the experimentally derived plots (upper limit, lower limit, and average of br and ar. Following is a step-by-step procedure to estimate the ultimate uplift capacity. Step 1. Step 2. Step 3. Step 4. Determine the representative value of the undrained cohesion, cu. Determine the critical embedment ratio using Eqs. (4.57) and (4.58). Determine the Df>B ratio for the foundation. If Df>B . (Df>B) cr, as determined in Step 2, it is a deep foundation. However, if Df>B # (Df>B) cr, it is a shallow foundation. 1.2 1.0 0.8 p l er Up 0.6 im it ge it era lim v A er w Lo 0.4 0.2 0 0 0.2 0.4 Figure 4.29 Plot of br versus ar 0.6 0.8 1.0 218 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases Step 5. For Df>B . (Df>B) cr Fc 5 F c* 5 7.56 1 1.44¢ B ≤ L Thus, Qu 5 Ab B 7.56 1 1.44¢ B ≤ Rcu 1 gDf r L (4.62) where A area of the foundation. Step 6. For Df>B # (Df>B) cr Qu 5 A(brF c* cu 1 gDf ) 5 Abbr B7.56 1 1.44¢ B ≤ R cu 1 gDf r (4.63) L The value of br can be obtained from the average curve of Figure 4.29. The procedure outlined above gives fairly good results for estimating the net ultimate uplift capacity of foundations and agrees reasonably well with the theoretical solution of Merifield et al. (2003). Example 4.12 Consider a circular foundation in sand. Given for the foundation: diameter, B 5 1.5 m and depth of embedment, Df 5 1.5 m. Given for the sand: unit weight, g 5 17.4 kN>m3 , and friction angle, fr 5 35°. Calculate the ultimate bearing capacity. Solution Df>B 5 1.5>1.5 5 1 and fr 5 35°. For circular foundation, (Df>B) cr 5 5. Hence, it is a shallow foundation. From Eq. (4.53) Fq 5 1 1 2B1 1 m¢ Df B ≤R¢ Df B ≤ Ku tan fr For fr 5 35°, m 0.25, and Ku 0.936 (Table 4.3). So Fq 5 1 1 231 1 (0.25) (1)4 (1) (0.936) (tan35) 5 2.638 So p Qu 5 FqgADf 5 (2.638) (17.4) B ¢ ≤ (1.5) 2 R (1.5) 5 121.7 kN 4 ■ Problems 219 Example 4.13 A rectangular foundation in a saturated clay measures 1.5 m 3 m. Given: Df 5 1.8 m, cu 5 52 kN>m2 , and g 5 18.9 kN>m3 . Estimate the ultimate uplift capacity. Solution From Eq. (4.57) ¢ Df B ≤ 5 0.107cu 1 2.5 5 (0.107) (52) 1 2.5 5 8.06 cr-square So use (Df>B) cr-square 5 7 . Again from Eq. (4.58), Df Df L 5 ¢ ≤ B0.73 1 0.27¢ ≤ R ¢ ≤ B cr-rectangular B cr-square B 5 7B0.73 1 0.27¢ Check: 1.55¢ Df B ≤ 3 ≤ R 5 8.89 1.5 5 (1.55) (7) 5 10.85 cr-square So use (Df>B) cr-rectangular 5 8.89 . The actual embedment ratio is Df>B 5 1.8>1.5 5 1.2. Hence, this is a shallow foundation. Df ar 5 ¢ B Df B 5 ≤ 1.2 5 0.135 8.89 cr Referring to the average curve of Figure 4.29, for ar 5 0.135, the magnitude of br 5 0.2. From Eq. (4.63), Qu 5 Abbr B7.56 1 1.44¢ B ≤ R cu 1 gDf r L 5 (1.5) (3) b (0.2) B7.56 1 1.44¢ 1.5 ≤ R (52) 1 (18.9) (1.8) r 5 540.6 kN 3 ■ Problems 4.1 Refer to Figure 4.2 and consider a rectangular foundation. Given: B 0.91 m, L 1.83 m, Df 0.91 m, H 0.61 m, 40°, c 0, and 18.08 kN/m3. Using a factor of safety of 4, determine the gross allowable load the foundation can carry. Use Eq. (4.3). 220 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 Repeat Problem 4.1 with the following data: B 1.5 m, L 1.5 m, Df 1 m, H 0.6 m, 35°, c 0, and 15 kN/m3. Use FS 3. Refer to Figure 4.2. Given: B L 1.75 m, Df 1 m, H 1.75 m, 17 kN/m3, c 0, and 30°. Using Eq. (4.6) and FS 4, determine the gross allowable load the foundation can carry. Refer to Figure 4.2. A square foundation measuring 1.22 m 1.22 m is supported by a saturated clay layer of limited depth underlain by a rock layer. Given that Df 0.91 m, H 0.61 m, cu 115 kN/m2, and 18.87 kN/m3, estimate the ultimate bearing capacity of the foundation. Refer to Figure 4.8. For a strip foundation in two-layered clay, given: • 1 18.08 kN/m3, c1 57.5 kN/m2, 1 0 • 2 17.29 kN/m3, c2 28.75 kN/m2, 2 0 • B 0.91 m, Df 0.61 m, H 0.61 m Find the gross allowable bearing capacity. Use a factor of safety of 3. Refer to Figure 4.8. For a strip foundation in two-layered clay, given: • B 0.92 m, L 1.22 m, Df 0.92 m, H 0.76 m • 1 17 kN/m3, 1 0, c1 72 kN/m2 • 2 17 kN/m3, 2 0, c2 43 kN/m2 Determine the gross ultimate bearing capacity. Refer to Figure 4.8. For a square foundation on layered sand, given: • B 1.5 m, Df 1.5 m, H 1 m • 1 18 kN/m3, 1 40°, c1 0 • 2 16.7 kN/m3, 2 32°, c2 0 Determine the net allowable load that the foundations can support. Use FS 4. Refer to Figure 4.11. For a rectangular foundation on layered sand, given: • B 1.22 m, L 1.83 m, H 0.61 m, Df 0.91 m • 1 15.41 kN/m3, 1 30°, c1 0 • 2 16.98 kN/m3, 2 38°, c2 0 Using a factor of safety of 4, determine the gross allowable load the foundation can carry. Two continuous shallow foundations are constructed alongside each other in a granular soil. Given, for the foundation: B 1.2 m, Df 1 m, and center-to-center spacing 2 m. The soil friction angle, 35°. Estimate the net allowable bearing capacity of the foundations. Use a factor of safety of FS 4 and a unit weight of soil g 5 16.8 kN>m3. A continuous foundation with a width of 1 m is located on a slope made of clay soil. Refer to Figure 4.14 and let Df 1 m, H 4 m, b 2 m, g 5 16.8 kN>m3, c 68 kN/m2 , 0, and 60°. a. Determine the allowable bearing capacity of the foundation. Let FS 3. b. Plot a graph of the ultimate bearing capacity qu if b is changed from 0 to 6 m. A continuous foundation is to be constructed near a slope made of granular soil (see Figure 4.14). If B 1.22 m, b 1.83 m, H 4.57 m, Df 1.22 m, b 30°, fr 5 40°, and g 5 17.29 kN>m3, estimate the ultimate bearing capacity of the foundation. Use Meyerhof’s solution. A square foundation in a sand deposit measures 1.22 m 1.22 m in plan. Given: Df 1.52 m, soil friction angle 5 35°, and unit weight of soil 5 17.6 kN>m3. Estimate the ultimate uplift capacity of the foundation. References 221 4.13 A foundation measuring 1.2 m 2.4 m in plan is constructed in a saturated clay. Given: depth of embedment of the foundation 2 m, unit weight of soil 18 kN>m3 , and undrained cohesion of clay 74 kN>m2 . Estimate the ultimate uplift capacity of the foundation. References BOWLES, J. E. (1996). Foundation Analysis and Design, 5th. edition, McGraw-Hill, New York. BUISMAN, A. S. K. (1940). Grondmechanica, Waltman, Delft, the Netherlands. CERATO, A. B., and LUTENEGGER, A. J. (2006). “Bearing Capacity of Square and Circular Footings on a Finite Layer of Granular Soil Underlain by a Rigid Base,” Journal of Geotechnical and Geoenvironmental Engineering, American Society of Civil Engineers, Vol. 132, No. 11, pp. 1496–1501. DAS, B. M. (1978). “Model Tests for Uplift Capacity of Foundations in Clay,” Soils and Foundations, Vol. 18, No. 2, pp. 17–24. DAS, B. M. (1980). “A Procedure for Estimation of Ultimate Uplift Capacity of Foundations in Clay,” Soils and Foundations, Vol. 20, No. 1, pp. 77–82. DAS, B. M., and JONES, A. D. (1982). “Uplift Capacity of Rectangular Foundations in Sand,” Transportation Research Record 884, National Research Council, Washington, D.C., pp. 54–58. DAS, B. M., and SEELEY, G. R. (1975). “Breakout Resistance of Horizontal Anchors,” Journal of Geotechnical Engineering Division, ASCE, Vol. 101, No. 9, pp. 999–1003. GRAHAM, J., ANDREWS, M., and SHIELDS, D. H. (1988). “Stress Characteristics for Shallow Footing on Cohesionless Slopes,” Canadian Geotechnical Journal, Vol. 25, No. 2, pp. 238–249. HUANG, C. C., and KANG, W. W. (2008). “Seismic Bearing Capacity of a Rigid Footing Adjacent to a Cohesionless Slope,” Soils and Foundations, Vol. 48, No. 5, pp. 641–651. LUNDGREN, H., and MORTENSEN, K. (1953). “Determination by the Theory of Plasticity on the Bearing Capacity of Continuous Footings on Sand,” Proceedings, Third International Conference on Soil Mechanics and Foundation Engineering, Zurich, Vol. 1, pp. 409 – 412. MANDEL, J., and SALENCON, J. (1972). “Force portante d’un sol sur une assise rigide (étude théorique),” Geotechnique, Vol. 22, No. 1, pp. 79–93. MERIFIELD, R. S., LYAMIN, A., and SLOAN, S. W. (2003). “Three Dimensional Lower Bound Solutions for the Stability of Plate Anchors in Clay,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vo. 129, No. 3, pp. 243–253. MEYERHOF, G. G. (1957). “The Ultimate Bearing Capacity of Foundations on Slopes,” Proceedings, Fourth International Conference on Soil Mechanics and Foundation Engineering, London, Vol. 1, pp. 384 –387. MEYERHOF, G. G. (1974). “Ultimate Bearing Capacity of Footings on Sand Layer Overlying Clay,” Canadian Geotechnical Journal, Vol. 11, No. 2, pp. 224– 229. MEYERHOF, G. G., and ADAMS, J. I. (1968). “The Ultimate Uplift Capacity of Foundations,” Canadian Geotechnical Journal, Vol. 5, No. 4, pp. 225–244. MEYERHOF, G. G., and CHAPLIN, T. K. (1953). “The Compression and Bearing Capacity of Cohesive Soils,” British Journal of Applied Physics, Vol. 4, pp. 20–26. MEYERHOF, G. G., and HANNA, A. M. (1978). “Ultimate Bearing Capacity of Foundations on Layered Soil under Inclined Load,” Canadian Geotechnical Journal, Vol. 15, No. 4, pp. 565– 572. 222 Chapter 4: Ultimate Bearing Capacity of Shallow Foundations: Special Cases PRANDTL, L. (1921). “Über die Eindringungsfestigkeit (Härte) plastischer Baustoffe und die Festigkeit von Schneiden,” Zeitschrift für angewandte Mathematik und Mechanik, Vol. 1, No. 1, pp. 15– 20. REISSNER, H. (1924). “Zum Erddruckproblem,” Proceedings, First International Congress of Applied Mechanics, Delft, the Netherlands, pp. 295–311. STAGG, K. G., and ZIENKIEWICZ, O. C. (1968). Rock Mechanics in Engineering Practice, John Wiley & Sons, New York. 5 5.1 Shallow Foundations: Allowable Bearing Capacity and Settlement Introduction It was mentioned in Chapter 3 that, in many cases, the allowable settlement of a shallow foundation may control the allowable bearing capacity. The allowable settlement itself may be controlled by local building codes. Thus, the allowable bearing capacity will be the smaller of the following two conditions: qu FS qall 5 d or qallowable settlement The settlement of a foundation can be divided into two major categories: (a) elastic, or immediate, settlement and (b) consolidation settlement. Immediate, or elastic, settlement of a foundation takes place during or immediately after the construction of the structure. Consolidation settlement occurs over time. Pore water is extruded from the void spaces of saturated clayey soils submerged in water. The total settlement of a foundation is the sum of the elastic settlement and the consolidation settlement. Consolidation settlement comprises two phases: primary and secondary. The fundamentals of primary consolidation settlement were explained in detail in Chapter 1. Secondary consolidation settlement occurs after the completion of primary consolidation caused by slippage and reorientation of soil particles under a sustained load. Primary consolidation settlement is more significant than secondary settlement in inorganic clays and silty soils. However, in organic soils, secondary consolidation settlement is more significant. For the calculation of foundation settlement (both elastic and consolidation), it is required that we estimate the vertical stress increase in the soil mass due to the net load applied on the foundation. Hence, this chapter is divided into the following three parts: 1. Procedure for calculation of vertical stress increase 2. Elastic settlement calculation 3. Consolidation settlement calculation 223 224 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Vertical Stress Increase in a Soil Mass Caused by Foundation Load 5.2 Stress Due to a Concentrated Load In 1885, Boussinesq developed the mathematical relationships for determining the normal and shear stresses at any point inside homogeneous, elastic, and isotropic mediums due to a concentrated point load located at the surface, as shown in Figure 5.1. According to his analysis, the vertical stress increase at point A caused by a point load of magnitude P is given by Ds 5 3P r 2 5>2 2pz B1 1 ¢ ≤ R z (5.1) 2 where r 5 "x2 1 y2 x, y, z 5 coordinates of the point A Note that Eq. (5.1) is not a function of Poisson’s ratio of the soil. 5.3 Stress Due to a Circularly Loaded Area The Boussinesq equation (5.1) can also be used to determine the vertical stress below the center of a flexible circularly loaded area, as shown in Figure 5.2. Let the radius of the loaded area be B>2, and let qo be the uniformly distributed load per unit area. To determine P x r y z A (x,y,z) Figure 5.1 Vertical stress at a point A caused by a point load on the surface 5.3 Stress Due to a Circularly Loaded Area 225 qo B/2 r r dr d qo z z A A Figure 5.2 Increase in pressure under a uniformly loaded flexible circular area the stress increase at a point A, located at a depth z below the center of the circular area, consider an elemental area on the circle. The load on this elemental area may be taken to be a point load and expressed as qor du dr. The stress increase at A caused by this load can be determined from Eq. (5.1) as 3(qor du dr) ds 5 (5.2) r 2 5>2 2pz B1 1 ¢ ≤ R z 2 The total increase in stress caused by the entire loaded area may be obtained by integrating Eq. (5.2), or u52p Ds 5 3ds 5 3 r5B>2 3 u50 5 qo d1 2 r50 3(qor du dr) r 2 5>2 2pz2 B1 1 ¢ ≤ R z 1 B 2 3>2 B1 1 ¢ ≤ R 2z t (5.3) Similar integrations could be performed to obtain the vertical stress increase at Ar, located a distance r from the center of the loaded area at a depth z (Ahlvin and Ulery, 1962). Table 5.1 gives the variation of Ds>qo with r> (B>2) and z> (B>2) [for 0 # r> (B>2) # 1]. Note that the variation of Ds>qo with depth at r> (B>2) 5 0 can be obtained from Eq. (5.3). 226 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Table 5.1 Variation of Ds>qo for a Uniformly Loaded Flexible Circular Area r , (B , 2) 5.4 z , (B , 2) 0 0.2 0.4 0.6 0.8 1.0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.5 2.0 2.5 3.0 4.0 1.000 0.999 0.992 0.976 0.949 0.911 0.864 0.811 0.756 0.701 0.646 0.546 0.424 0.286 0.200 0.146 0.087 1.000 0.999 0.991 0.973 0.943 0.902 0.852 0.798 0.743 0.688 0.633 0.535 0.416 0.286 0.197 0.145 0.086 1.000 0.998 0.987 0.963 0.920 0.869 0.814 0.756 0.699 0.644 0.591 0.501 0.392 0.268 0.191 0.141 0.085 1.000 0.996 0.970 0.922 0.860 0.796 0.732 0.674 0.619 0.570 0.525 0.447 0.355 0.248 0.180 0.135 0.082 1.000 0.976 0.890 0.793 0.712 0.646 0.591 0.545 0.504 0.467 0.434 0.377 0.308 0.224 0.167 0.127 0.080 1.000 0.484 0.468 0.451 0.435 0.417 0.400 0.367 0.366 0.348 0.332 0.300 0.256 0.196 0.151 0.118 0.075 Stress below a Rectangular Area The integration technique of Boussinesq’s equation also allows the vertical stress at any point A below the corner of a flexible rectangular loaded area to be evaluated. (See Figure 5.3.) To do so, consider an elementary area dA 5 dx dy on the flexible loaded area. If the load per unit area is qo , the total load on the elemental area is dP 5 qo dx dy (5.4) This elemental load, dP, may be treated as a point load. The increase in vertical stress at point A caused by dP may be evaluated by using Eq. (5.1). Note, however, the need to substitute dP 5 qo dx dy for P and x2 1 y2 for r2 in that equation. Thus, The stress increase at A caused by dP 5 3qo (dx dy)z3 2p(x2 1 y2 1 z2 ) 5>2 The total stress increase Ds caused by the entire loaded area at point A may now be obtained by integrating the preceding equation: L Ds 5 3 B 3qo (dx dy)z3 3 2 y50 x50 2p(x 1 y2 1 z2 ) 5>2 5 qoI (5.5) 227 5.4 Stress below a Rectangular Area x qo B dx dy y L z Figure 5.3 Determination of stress below the corner of a flexible rectangular loaded area A Here, I 5 influence factor 5 2mn"m2 1 n2 1 1 # m2 1 n2 1 2 1 ¢ 2 4p m 1 n2 1 m2n2 1 1 m2 1 n2 1 1 1 tan21 2mn"m2 1 n2 1 1 ≤ m2 1 n2 1 1 2 m2n2 (5.6) where m5 B z (5.7) n5 L z (5.8) and The variations of the influence values with m and n are given in Table 5.2. The stress increase at any point below a rectangular loaded area can also be found by using Eq. (5.5) in conjunction with Figure 5.4. To determine the stress at a depth z below point O, divide the loaded area into four rectangles, with O the corner common to each. Then use Eq. (5.5) to calculate the increase in stress at a depth z below O caused by each rectangular area. The total stress increase caused by the entire loaded area may now be expressed as Ds 5 qo (I1 1 I2 1 I3 1 I4 ) (5.9) where I1 , I2 , I3 , and I4 5 the influence values of rectangles 1, 2, 3, and 4, respectively. In most cases, the vertical stress below the center of a rectangular area is of importance. This can be given by the relationship Ds 5 qoIc (5.10) 228 0.1 0.00470 0.00917 0.01323 0.01678 0.01978 0.02223 0.02420 0.02576 0.02698 0.02794 0.02926 0.03007 0.03058 0.03090 0.03111 0.03138 0.03150 0.03158 0.03160 0.03161 0.03162 0.03162 0.03162 m 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 4.0 5.0 6.0 8.0 10.0 ` 0.06201 0.06202 0.06202 0.06202 0.06155 0.06178 0.06194 0.06199 0.05733 0.05894 0.05994 0.06058 0.06100 0.04348 0.04735 0.05042 0.05283 0.05471 0.00917 0.01790 0.02585 0.03280 0.03866 0.2 0.09017 0.09018 0.09019 0.09019 0.08948 0.08982 0.09007 0.09014 0.08323 0.08561 0.08709 0.08804 0.08867 0.06294 0.06858 0.07308 0.07661 0.07938 0.01323 0.02585 0.03735 0.04742 0.05593 0.3 0.11541 0.11543 0.11544 0.11544 0.11450 0.11495 0.11527 0.11537 0.10631 0.10941 0.11135 0.11260 0.11342 0.08009 0.08734 0.09314 0.09770 0.10129 0.01678 0.03280 0.04742 0.06024 0.07111 0.4 Table 5.2 Variation of Influence Value I [Eq. (5.6)]a 0.13741 0.13744 0.13745 0.13745 0.13628 0.13684 0.13724 0.13737 0.12626 0.13003 0.13241 0.13395 0.13496 0.09473 0.10340 0.11035 0.11584 0.12018 0.01978 0.03866 0.05593 0.07111 0.08403 0.5 0.15617 0.15621 0.15622 0.15623 0.15483 0.15550 0.15598 0.15612 0.14309 0.14749 0.15028 0.15207 0.15326 0.10688 0.11679 0.12474 0.13105 0.13605 0.02223 0.04348 0.06294 0.08009 0.09473 0.6 n 0.17191 0.17195 0.17196 0.17197 0.17036 0.17113 0.17168 0.17185 0.15703 0.16199 0.16515 0.16720 0.16856 0.11679 0.12772 0.13653 0.14356 0.14914 0.02420 0.04735 0.06858 0.08734 0.10340 0.7 0.18496 0.18500 0.18502 0.18502 0.18321 0.18407 0.18469 0.18488 0.16843 0.17389 0.17739 0.17967 0.18119 0.12474 0.13653 0.14607 0.15371 0.15978 0.02576 0.05042 0.07308 0.09314 0.11035 0.8 0.19569 0.19574 0.19576 0.19577 0.19375 0.19470 0.19540 0.19561 0.17766 0.18357 0.18737 0.18986 0.19152 0.13105 0.14356 0.15371 0.16185 0.16835 0.02698 0.05283 0.07661 0.09770 0.11584 0.9 0.20449 0.20455 0.20457 0.20458 0.20236 0.20341 0.20417 0.20440 0.18508 0.19139 0.19546 0.19814 0.19994 0.13605 0.14914 0.15978 0.16835 0.17522 0.02794 0.05471 0.07938 0.10129 0.12018 1.0 0.21760 0.21767 0.21769 0.21770 0.21512 0.21633 0.21722 0.21749 0.19584 0.20278 0.20731 0.21032 0.21235 0.14309 0.15703 0.16843 0.17766 0.18508 0.02926 0.05733 0.08323 0.10631 0.12626 1.2 0.22644 0.22652 0.22654 0.22656 0.22364 0.22499 0.22600 0.22632 0.20278 0.21020 0.21510 0.21836 0.22058 0.14749 0.16199 0.17389 0.18357 0.19139 0.03007 0.05894 0.08561 0.10941 0.13003 1.4 229 a 0.03058 0.05994 0.08709 0.11135 0.13241 0.15028 0.16515 0.17739 0.18737 0.19546 0.20731 0.21510 0.22025 0.22372 0.22610 0.22940 0.23088 0.23200 0.23236 0.23249 0.23258 0.23261 0.23263 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 4.0 5.0 6.0 8.0 10.0 ` After Newmark, 1935. 1.6 m 1.8 0.23671 0.23681 0.23684 0.23686 0.23334 0.23495 0.23617 0.23656 0.21032 0.21836 0.22372 0.22736 0.22986 0.15207 0.16720 0.17967 0.18986 0.19814 0.03090 0.06058 0.08804 0.11260 0.13395 Table 5.2 (Continued) 0.23970 0.23981 0.23985 0.23987 0.23614 0.23782 0.23912 0.23954 0.21235 0.22058 0.22610 0.22986 0.23247 0.15326 0.16856 0.18119 0.19152 0.19994 0.03111 0.06100 0.08867 0.11342 0.13496 2.0 0.24412 0.24425 0.24429 0.24432 0.24010 0.24196 0.24344 0.24392 0.21512 0.22364 0.22940 0.23334 0.23614 0.15483 0.17036 0.18321 0.19375 0.20236 0.03138 0.06155 0.08948 0.11450 0.13628 2.5 0.24630 0.24646 0.24650 0.24654 0.24196 0.24394 0.24554 0.24608 0.21633 0.22499 0.23088 0.23495 0.23782 0.15550 0.17113 0.18407 0.19470 0.20341 0.03150 0.06178 0.08982 0.11495 0.13684 3.0 0.24817 0.24836 0.24842 0.24846 0.24344 0.24554 0.24729 0.24791 0.21722 0.22600 0.23200 0.23617 0.23912 0.15598 0.17168 0.18469 0.19540 0.20417 0.03158 0.06194 0.09007 0.11527 0.13724 4.0 n 0.24885 0.24907 0.24914 0.24919 0.24392 0.24608 0.24791 0.24857 0.21749 0.22632 0.23236 0.23656 0.23954 0.15612 0.17185 0.18488 0.19561 0.20440 0.03160 0.06199 0.09014 0.11537 0.13737 5.0 0.24916 0.24939 0.24946 0.24952 0.24412 0.24630 0.24817 0.24885 0.21760 0.22644 0.23249 0.23671 0.23970 0.15617 0.17191 0.18496 0.19569 0.20449 0.03161 0.06201 0.09017 0.11541 0.13741 6.0 0.24939 0.24964 0.24973 0.24980 0.24425 0.24646 0.24836 0.24907 0.21767 0.22652 0.23258 0.23681 0.23981 0.15621 0.17195 0.18500 0.19574 0.20455 0.03162 0.06202 0.09018 0.11543 0.13744 8.0 0.24946 0.24973 0.24981 0.24989 0.24429 0.24650 0.24842 0.24914 0.21769 0.22654 0.23261 0.23684 0.23985 0.15622 0.17196 0.18502 0.19576 0.20457 0.03162 0.06202 0.09019 0.11544 0.13745 10.0 0.24952 0.24980 0.24989 0.25000 0.24432 0.24654 0.24846 0.24919 0.21770 0.22656 0.23263 0.23686 0.23987 0.15623 0.17197 0.18502 0.19577 0.20458 0.03162 0.06202 0.09019 0.11544 0.13745 ⴥ 230 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement B(1) 1 3 O B(2) 2 4 L(1) L(2) Figure 5.4 Stress below any point of a loaded flexible rectangular area where Ic 5 m1n1 1 1 m21 1 2n21 2 B p "1 1 m2 1 n2 (1 1 n21 ) (m21 1 n21 ) 1 1 1 sin21 m1 5 n1 5 L B "m21 m1 1 n21"1 1 n21 R (5.11) (5.12) z (5.13) B ¢ ≤ 2 The variation of Ic with m1 and n1 is given in Table 5.3. Table 5.3 Variation of Ic with m1 and n1 m1 ni 1 2 3 4 5 6 7 8 9 10 0.20 0.40 0.60 0.80 1.00 1.20 1.40 1.60 1.80 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 0.994 0.960 0.892 0.800 0.701 0.606 0.522 0.449 0.388 0.336 0.179 0.108 0.072 0.051 0.038 0.029 0.023 0.019 0.997 0.976 0.932 0.870 0.800 0.727 0.658 0.593 0.534 0.481 0.293 0.190 0.131 0.095 0.072 0.056 0.045 0.037 0.997 0.977 0.936 0.878 0.814 0.748 0.685 0.627 0.573 0.525 0.348 0.241 0.174 0.130 0.100 0.079 0.064 0.053 0.997 0.977 0.936 0.880 0.817 0.753 0.692 0.636 0.585 0.540 0.373 0.269 0.202 0.155 0.122 0.098 0.081 0.067 0.997 0.977 0.937 0.881 0.818 0.754 0.694 0.639 0.590 0.545 0.384 0.285 0.219 0.172 0.139 0.113 0.094 0.079 0.997 0.977 0.937 0.881 0.818 0.755 0.695 0.640 0.591 0.547 0.389 0.293 0.229 0.184 0.150 0.125 0.105 0.089 0.997 0.977 0.937 0.881 0.818 0.755 0.695 0.641 0.592 0.548 0.392 0.298 0.236 0.192 0.158 0.133 0.113 0.097 0.997 0.977 0.937 0.881 0.818 0.755 0.696 0.641 0.592 0.549 0.393 0.301 0.240 0.197 0.164 0.139 0.119 0.103 0.997 0.977 0.937 0.881 0.818 0.755 0.696 0.641 0.593 0.549 0.394 0.302 0.242 0.200 0.168 0.144 0.124 0.108 0.997 0.977 0.937 0.881 0.818 0.755 0.696 0.642 0.593 0.549 0.395 0.303 0.244 0.202 0.171 0.147 0.128 0.112 5.4 Stress below a Rectangular Area 231 qo Foundation B L 2 vertical to 1 horizontal B 2 vertical to 1 horizontal z Bz Figure 5.5 2:1 method of finding stress increase under a foundation Foundation engineers often use an approximate method to determine the increase in stress with depth caused by the construction of a foundation. The method is referred to as the 2:1 method. (See Figure 5.5.) According to this method, the increase in stress at depth z is Ds 5 qo 3 B 3 L (B 1 z) (L 1 z) (5.14) ` Note that Eq. (5.14) is based on the assumption that the stress from the foundation spreads out along lines with a vertical-to-horizontal slope of 2:1. Example 5.1 A flexible rectangular area measures 2.5 m 3 5 m in plan. It supports a load of 150 kN> m2. Determine the vertical stress increase due to the load at a depth of 6.25 m below the center of the rectangular area. Solution Refer to Figure 5.4. For this case, 2.5 5 1.25 m 2 5 L1 5 L2 5 5 2.5 m 2 B1 5 B2 5 232 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement From Eqs. (5.7) and (5.8), B1 B2 1.25 5 5 5 0.2 z z 6.25 L1 L2 2.5 n5 5 5 5 0.4 z z 6.25 m5 From Table 5.2, for m 5 0.2 and n 5 0.4, the value of I 5 0.0328. Thus, Ds 5 qo (4I) 5 (150) (4) (0.0328) 5 19.68 kN , m2 Alternate Solution From Eq. (5.10), Ds 5 qoIc L 5 m1 5 5 52 B 2.5 z 6.25 n1 5 5 55 B 2.5 a b a b 2 2 From Table 5.3, for m1 5 2 and n1 5 5, the value of Ic 5 0.131. Thus, Ds 5 (150) (0.131) 5 19.65 kN , m2 5.5 ■ Average Vertical Stress Increase Due to a Rectangularly Loaded Area In Section 5.4, the vertical stress increase below the corner of a uniformly loaded rectangular area was given as Ds 5 qoI In many cases, one must find the average stress increase, Dsav , below the corner of a uniformly loaded rectangular area with limits of z 5 0 to z 5 H, as shown in Figure 5.6. This can be evaluated as H Dsav 5 1 (qoI) dz 5 qoIa H 30 (5.15) where Ia 5 f(m2, n2 ) B m2 5 H (5.16) (5.17) 5.5 Average Vertical Stress Increase Due to a Rectangularly Loaded Area 233 qo /unit area av z Section of loaded area dz H A (a) z (c) B Plan of loaded area L A (b) Figure 5.6 Average vertical stress increase due to a rectangularly loaded flexible area and n2 5 L H (5.18) The variation of Ia with m2 and n2 is shown in Figure 5.7, as proposed by Griffiths (1984). In estimating the consolidation settlement under a foundation, it may be required to determine the average vertical stress increase in only a given layer—that is, between z 5 H1 and z 5 H2 , as shown in Figure 5.8. This can be done as (Griffiths, 1984) Dsav(H2>H1) 5 qo B H2Ia(H2) 2 H1Ia(H1) H2 2 H1 R where Dsav(H2>H1) 5 average stress increase immediately below the corner of a uniformly loaded rectangular area between depths z H1 and z H2 Ia(H2) 5 Ia for z 5 0 to z 5 H2 5 f¢ m2 5 B L , n2 5 ≤ H2 H2 Ia(H1) 5 Ia for z 5 0 to z 5 H1 5 f¢m2 5 B L , n2 5 ≤ H1 H1 (5.19) 234 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement 0.26 n2 2.0 1.0 0.8 0.6 0.5 0.4 0.24 0.22 0.20 0.18 0.3 0.16 0.14 Ia 0.2 0.12 0.1 0.1 0.08 0.06 0.04 0.02 0.00 0.1 0.2 0.3 0.4 0.50.6 0.8 1.0 2 3 4 5 6 7 8 9 10 m2 Figure 5.7 Griffiths’ influence factor Ia qo /unit area Section H1 H2 av(H2/H1) z A A z B L Plan A, A Figure 5.8 Average pressure increase between z 5 H1 and z 5 H2 below the corner of a uniformly loaded rectangular area 5.5 Average Vertical Stress Increase Due to a Rectangularly Loaded Area 235 Example 5.2 Refer to Figure 5.9. Determine the average stress increase below the center of the loaded area between z 5 3 m to z 5 5 m (that is, between points A and A9). Solution Refer to Figure 5.9. The loaded area can be divided into four rectangular areas, each measuring 1.5 m 3 1.5 m (L 3 B). Using Eq. (5.19), the average stress increase (between the required depths) below the corner of each rectangular area can be given as Dsav (H2>H1) 5 qo B H2Ia(H2) 2 H1Ia(H1) H2 2 H1 R 5 100 B (5)Ia(H2) 2 (3)Ia(H1) 523 R For Ia(H2): B 1.5 5 5 0.3 H2 5 L 1.5 n2 5 5 5 0.3 H2 5 m2 5 Referring to Figure 5.7, for m2 5 0.3 and n2 5 0.3, Ia(H2) 5 0.126. For Ia(H1): B 1.5 5 5 0.5 H1 3 L 1.5 n2 5 5 5 0.5 H1 3 m2 5 qo 100 kN/m2 1.5 m 1.5 m z 3m 5m Section A A B A, A 3m Plan L 3m Figure 5.9 Determination of average increase in stress below a rectangular area 236 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Referring to Figure 5.7, Ia(H1) 5 0.175, so Dsav (H2>H1) 5 100 B (5) (0.126) 2 (3) (0.175) R 5 5.25 kN , m2 523 The stress increase between z 5 3 m to z 5 5 m below the center of the loaded area is equal to 4Dsav (H2>H1) 5 (4) (5.25) 5 21 kN , m2 5.6 ■ Stress Increase under an Embankment Figure 5.10 shows the cross section of an embankment of height H. For this two-dimensional loading condition, the vertical stress increase may be expressed as Ds 5 qo B1 1 B2 B1 B¢ ≤ (a1 1 a2 ) 2 (a ) R p B2 B2 2 (5.20) where qo 5 gH g 5 unit weight of the embankment soil H 5 height of the embankment a1 5 tan21 ¢ B1 1 B2 B1 ≤ 2 tan21 ¢ ≤ z z (5.21) a2 5 tan21 ¢ B1 ≤ z (5.22) (Note that a1 and a2 are in radians.) B2 B1 qo H H 1 2 z Figure 5.10 Embankment loading 5.6 Stress Increase under an Embankment 0.50 237 3.0 2.0 1.6 1.4 1.2 1.0 0.9 0.8 0.45 0.40 0.7 0.35 0.6 0.30 I 0.5 0.4 0.25 0.3 0.20 0.2 0.15 0.10 0.1 0.05 B1 /z = 0 0 0.01 0.1 1.0 B2 /z Figure 5.11 Influence value Ir for embankment loading (After Osterberg, 1957) (Osterberg, J. O. (1957). “Influence Values for Vertical Stresses in Semi-Infinite Mass Due to Embankment Loading,” Proceedings, Fourth International Conference on Soil Mechanics and 10 Foundation Engineering, London, Vol. 1. pp. 393–396. With permission from ASCE.) For a detailed derivation of Eq. (5.20), see Das (2008). A simplified form of the equation is Ds 5 qoIr (5.23) where Ir 5 a function of B1>z and B2>z. The variation of Ir with B1>z and B2>z is shown in Figure 5.11. An application of this diagram is given in Example 5.3. Example 5.3 An embankment is shown in Figure 5.12a. Determine the stress increase under the embankment at points A 1 and A 2 . Solution We have gH 5 (17.5) (7) 5 122.5 kN>m2 238 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Stress Increase at A1 The left side of Figure 5.12b indicates that B1 5 2.5 m and B2 5 14 m, so B1 2.5 5 0.5 5 z 5 and B2 14 5 2.8 5 z 5 14 m 5m 14 m H7m 17.5 kN/m3 5m 11.5 m 16.5 m 5m 5m A2 A1 (a) 2.5 m 2.5 m 14 m 14 m qo qo 122.5 122.5 kN/m2 kN/m2 5m (1) (2) A2 A1 (b) 14 m 5m 14 m qo (7 m) qo (2.5 m) 3 (17.5 kN/m3) (17.5 kN/m ) 2 122.5 kN/m2 43.75 kN/m 5m (1) A2 (2) A2 qo (4.5 m) (17.5 kN/m3) 78.75 kN/m2 9m (3) (c) Figure 5.12 Stress increase due to embankment loading A2 5.6 Stress Increase under an Embankment 239 According to Figure 5.11, in this case Ir 5 0.445. Because the two sides in Figure 5.12b are symmetrical, the value of Ir for the right side will also be 0.445, so Ds 5 Ds(1) 1 Ds(2) 5 qo 3I (left r side) 1 I (right r side) 4 5 122.530.445 1 0.4454 5 109.03 kN , m2 Stress Increase at A2 In Figure 5.12c, for the left side, B2 5 5 m and B1 5 0, so B2 5 5 51 z 5 and B1 0 5 50 z 5 According to Figure 5.11, for these values of B2>z and B1>z, I r 5 0.24; hence, Ds(1) 5 43.75(0.24) 5 10.5 kN>m2 For the middle section, B2 14 5 5 2.8 z 5 and B1 14 5 5 2.8 z 5 Thus, I r 5 0.495, so Ds(2) 5 0.495(122.5) 5 60.64 kN>m2 For the right side, B2 9 5 5 1.8 z 5 B1 0 5 50 z 5 and I r 5 0.335, so Ds(3) 5 (78.75) (0.335) 5 26.38 kN>m2 The total stress increase at point A 2 is Ds 5 Ds(1) 1 Ds(2) 2 Ds(3) 5 10.5 1 60.64 2 26.38 5 44.76 kN , m2 ■ 240 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement 5.7 Westergaard’s Solution for Vertical Stress Due to a Point Load Boussinesq’s solution for stress distribution due to a point load was presented in Section 5.2. The stress distribution due to various types of loading discussed in Sections 5.3 through 5.6 is based on integration of Boussinesq’s solution. Westergaard (1938) has proposed a solution for the determination of the vertical stress due to a point load P in an elastic solid medium in which there exist alternating layers with thin rigid reinforcements (Figure 5.13a). This type of assumption may be an idealization of a clay layer with thin seams of sand. For such an assumption, the vertical stress increase at a point A (Figure 5.13b) can be given as Ds 5 Ph 3>2 1 d 2 2pz h 1 (r>z) 2 c 2 (5.24) P Thin rigid reinforcement s = Poisson s ratio of soil between the rigid layers (a) P x r z y A z (b) Figure 5.13 Westergaard’s solution for vertical stress due to a point load 5.8 Stress Distribution for Westergaard Material 241 Table 5.4 Variation of I1 [Eq. (5.27)]. I1 r/z s ⴝ 0 s ⴝ 0.2 s ⴝ 0.4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.0 2.5 3.0 4.0 5.0 0.3183 0.3090 0.2836 0.2483 0.2099 0.1733 0.1411 0.1143 0.0925 0.0751 0.0613 0.0247 0.0118 0.0064 0.0038 0.0017 0.0009 0.4244 0.4080 0.3646 0.3074 0.2491 0.1973 0.1547 0.1212 0.0953 0.0756 0.0605 0.0229 0.0107 0.0057 0.0034 0.0015 0.0008 0.9550 0.8750 0.6916 0.4997 0.3480 0.2416 0.1700 0.1221 0.0897 0.0673 0.0516 0.0173 0.0076 0.0040 0.0023 0.0010 0.0005 where 1 2 2ms Ä 2 2 2ms s Poisson’s ratio of the solid between the rigid reinforcements r 5 !x2 1 y2 Equation (5.24) can be rewritten as h5 Ds 5 a P bI1 z2 (5.25) (5.26) where I1 5 23>2 1 r 2 c a b 1 1d 2ph2 hz (5.27) Table 5.4 gives the variation of I1 with s. 5.8 Stress Distribution for Westergaard Material Stress Due to a Circularly Loaded Area Referring to Figure 5.2, if the circular area is located on a Westergaard-type material, the increase in vertical stress, , at a point located at a depth z immediately below the center of the area can be given as 242 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Ds 5 qo μ 1 2 h B 2 1>2 ch 1 a b d 2z 2 ∂ (5.28) The term has been defined in Eq. (5.25). The variations of /qo with B/2z and s 0 are given in Table 5.5. Stress Due to a Uniformly Loaded Flexible Rectangular Area Refer to Figure 5.3. If the flexible rectangular area is located on a Westergaard-type material, the stress increase at point A can be given as Ds 5 qo 1 1 1 ccot21 h2 a 2 1 2 b 1 h4 a 2 2 b d 2p Ä m n mn where m5 B z n5 L z Table 5.5 Variation of /qo with B/2z and s 0 [Eq. (5.28)] B/2z ⌬/qo 0.00 0.25 0.33 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 0.0 0.0572 0.0938 0.1835 0.3140 0.4227 0.5076 0.5736 0.6254 0.6667 0.7002 0.7278 0.7510 0.7706 0.8259 0.8600 0.8830 0.8995 0.9120 0.9217 0.9295 (5.29a) 5.9 Elastic Settlement of Foundations on Saturated Clay (s 0.5) 243 Table 5.6. Variation of Iw with m and n (s 0). n m 0.1 0.2 0.4 0.5 0.6 1.0 2.0 5.0 10.0 0.1 0.2 0.4 0.5 0.6 1.0 2.0 5.0 10.0 0.0031 0.0061 0.0110 0.0129 0.0144 0.0183 0.0211 0.0221 0.0223 0.0061 0.0118 0.0214 0.0251 0.0282 0.0357 0.0413 0.0435 0.0438 0.0110 0.0214 0.0390 0.0459 0.0516 0.0658 0.0768 0.0811 0.0817 0.0129 0.0251 0.0459 0.0541 0.0610 0.0781 0.0916 0.0969 0.0977 0.0144 0.0282 0.0516 0.0610 0.0687 0.0886 0.1044 0.1107 0.1117 0.0182 0.0357 0.0658 0.0781 0.0886 0.1161 0.1398 0.1499 0.1515 0.0211 0.0413 0.0768 0.0916 0.1044 0.1398 0.1743 0.1916 0.1948 0.0211 0.0434 0.0811 0.0969 0.1107 0.1491 0.1916 0.2184 0.2250 0.0223 0.0438 0.0847 0.0977 0.1117 0.1515 0.1948 0.2250 0.2341 or Ds 1 1 1 1 5 ccot 21 h2 a 2 1 2 b 1 h4 a 2 2 b d 5 Iw qo 2p Å m n mn (5.29b) Table 5.6 gives the variation of Iw with m and n (for s 0). Example 5.4 Solve Example 5.1 using Eq. (5.29). Assume s 0. Solution From Example 5.1 m 0.2 n 0.4 qo(4Iw) From Table 5.6, for m 0.2 and n 0.4, the value of Iw ⬇ 0.0214. So (150)(4 0.0214) 12.84 kN/m2 ■ Elastic Settlement 5.9 Elastic Settlement of Foundations on Saturated Clay (s 0.5) Janbu et al. (1956) proposed an equation for evaluating the average settlement of flexible foundations on saturated clay soils (Poisson’s ratio, s 0.5). For the notation used in Figure 5.14, this equation is 244 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Se 5 A1A2 qoB Es (5.30) where A1 is a function of H/B and L/B and A2 is a function of Df /B. Christian and Carrier (1978) modified the values of A1 and A2 to some extent as presented in Figure 5.14. qo B Df H 1.0 A2 0.9 0.8 0 5 10 Df /B 15 20 2.0 L /B L /B 10 1.5 5 A1 1.0 2 Square Circle 0.5 0 0.1 1 10 H/B 100 1000 Figure 5.14 Values of A1 and A2 for elastic settlement calculation—Eq. (5.30) (After Christian and Carrier, 1978) (Christian, J. T. and and Carrier, W. D. (1978). “Janbu, Bjerrum and Kjaernsli's chart reinterpreted,” Canadian Geotechnical Journal, Vol. 15, pp. 123–128. © 2008 NRC Canada or its licensors. Reproduced with permission.) 5.10 Settlement Based on the Theory of Elasticity 245 Table 5.7 Range of for Clay [Eq. (5.31)]a  Plasticity Index OCR 1 OCR 2 OCR 3 OCR 4 OCR 5 30 30 to 50 50 1500–600 600–300 300–150 1380–500 550–270 270–120 1200–580 580–220 220–100 950–380 380–180 180–90 730–300 300–150 150–75 a Interpolated from Duncan and Buchignani, (1976) The modulus of elasticity (Es) for clays can, in general, be given as Es cu (5.31) where cu undrained shear strength. The parameter is primarily a function of the plasticity index and overconsolidation ratio. Table 5.7 provides a general range for based on that proposed by Duncan and Buchignani (1976). In any case, proper judgment should be used in selecting the magnitude of . 5.10 Settlement Based on the Theory of Elasticity The elastic settlement of a shallow foundation can be estimated by using the theory of elasticity. From Hooke’s law, as applied to Figure 5.15, we obtain H H 1 Se 5 3 ezdz 5 (Dsz 2 ms Dsx 2 ms Dsy )dz Es 30 0 Load qo /unit area z y H dz x y z Incompressible layer Figure 5.15 Elastic settlement of shallow foundation (5.32) 246 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement where Se 5 elastic settlement Es 5 modulus of elasticity of soil H 5 thickness of the soil layer ms 5 Poisson’s ratio of the soil Dsx , Dsy , Dsz 5 stress increase due to the net applied foundation load in the x, y, and z directions, respectively Theoretically, if the foundation is perfectly flexible (see Figure 5.16 and Bowles, 1987), the settlement may be expressed as Se 5 qo (aBr ) 1 2 m2s IsIf Es (5.33) where qo 5 net applied pressure on the foundation ms 5 Poisson’s ratio of soil Es 5 average modulus of elasticity of the soil under the foundation, measured from z 5 0 to about z 5 5B B r 5 B>2 for center of foundation 5 B for corner of foundation Is 5 shape factor (Steinbrenner, 1934) 5 F1 1 F1 5 Rigid foundation settlement (5.34) 1 (A 1 A 1 ) p 0 Foundation B L z 1 2 2ms F 1 2 ms 2 qo (5.35) Df Flexible foundation settlement H s Poisson s ratio Es Modulus of elasticity Soil Rock Figure 5.16 Elastic settlement of flexible and rigid foundations 5.10 Settlement Based on the Theory of Elasticity F2 5 nr tan21A 2 2p A 0 5 mrln (5.36) ¢1 1 "mr2 1 1≤"mr2 1 nr2 A2 5 (5.37) mr ¢1 1 "mr 1 nr 1 1≤ 2 A 1 5 ln 247 2 ¢ mr 1 "mr2 1 1≤"1 1 nr2 (5.38) mr 1 "mr2 1 nr2 1 1 mr nr"mr2 1 nr2 1 1 If 5 depth factor (Fox, 1948) 5 f¢ (5.39) Df B , ms , and L ≤ B (5.40) a 5 a factor that depends on the location on the foundation where settlement is being calculated To calculate settlement at the center of the foundation, we use a54 L mr 5 B and H nr 5 ¢ B ≤ 2 To calculate settlement at a corner of the foundation, a51 L mr 5 B and nr 5 H B The variations of F1 and F2 [see Eqs. (5.35) and (5.36)] with mr and nr are given in Tables 5.8 and 5.9. Also, the variation of If with Df>B (for ms 5 0.3, 0.4, and 0.5) is given in Table 5.10. These values are also given in more detailed form by Bowles (1987). The elastic settlement of a rigid foundation can be estimated as Se(rigid) < 0.93Se(flexible, center) (5.41) 248 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Table 5.8 Variation of F1 with m9 and n9 mⴕ nⴕ 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00 20.00 50.00 100.00 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 3.5 4.0 0.014 0.049 0.095 0.142 0.186 0.224 0.257 0.285 0.309 0.330 0.348 0.363 0.376 0.388 0.399 0.408 0.417 0.424 0.431 0.437 0.443 0.448 0.453 0.457 0.461 0.465 0.468 0.471 0.474 0.477 0.480 0.482 0.485 0.487 0.489 0.491 0.493 0.495 0.496 0.498 0.529 0.548 0.555 0.013 0.046 0.090 0.138 0.183 0.224 0.259 0.290 0.317 0.341 0.361 0.379 0.394 0.408 0.420 0.431 0.440 0.450 0.458 0.465 0.472 0.478 0.483 0.489 0.493 0.498 0.502 0.506 0.509 0.513 0.516 0.519 0.522 0.524 0.527 0.529 0.531 0.533 0.536 0.537 0.575 0.598 0.605 0.012 0.044 0.087 0.134 0.179 0.222 0.259 0.292 0.321 0.347 0.369 0.389 0.406 0.422 0.436 0.448 0.458 0.469 0.478 0.487 0.494 0.501 0.508 0.514 0.519 0.524 0.529 0.533 0.538 0.541 0.545 0.549 0.552 0.555 0.558 0.560 0.563 0.565 0.568 0.570 0.614 0.640 0.649 0.011 0.042 0.084 0.130 0.176 0.219 0.258 0.292 0.323 0.350 0.374 0.396 0.415 0.431 0.447 0.460 0.472 0.484 0.494 0.503 0.512 0.520 0.527 0.534 0.540 0.546 0.551 0.556 0.561 0.565 0.569 0.573 0.577 0.580 0.583 0.587 0.589 0.592 0.595 0.597 0.647 0.678 0.688 0.011 0.041 0.082 0.127 0.173 0.216 0.255 0.291 0.323 0.351 0.377 0.400 0.420 0.438 0.454 0.469 0.481 0.495 0.506 0.516 0.526 0.534 0.542 0.550 0.557 0.563 0.569 0.575 0.580 0.585 0.589 0.594 0.598 0.601 0.605 0.609 0.612 0.615 0.618 0.621 0.677 0.711 0.722 0.011 0.040 0.080 0.125 0.170 0.213 0.253 0.289 0.322 0.351 0.378 0.402 0.423 0.442 0.460 0.476 0.484 0.503 0.515 0.526 0.537 0.546 0.555 0.563 0.570 0.577 0.584 0.590 0.596 0.601 0.606 0.611 0.615 0.619 0.623 0.627 0.631 0.634 0.638 0.641 0.702 0.740 0.753 0.010 0.038 0.077 0.121 0.165 0.207 0.247 0.284 0.317 0.348 0.377 0.402 0.426 0.447 0.467 0.484 0.495 0.516 0.530 0.543 0.555 0.566 0.576 0.585 0.594 0.603 0.610 0.618 0.625 0.631 0.637 0.643 0.648 0.653 0.658 0.663 0.667 0.671 0.675 0.679 0.756 0.803 0.819 0.010 0.038 0.076 0.118 0.161 0.203 0.242 0.279 0.313 0.344 0.373 0.400 0.424 0.447 0.458 0.487 0.514 0.521 0.536 0.551 0.564 0.576 0.588 0.598 0.609 0.618 0.627 0.635 0.643 0.650 0.658 0.664 0.670 0.676 0.682 0.687 0.693 0.697 0.702 0.707 0.797 0.853 0.872 0.010 0.037 0.074 0.116 0.158 0.199 0.238 0.275 0.308 0.340 0.369 0.396 0.421 0.444 0.466 0.486 0.515 0.522 0.539 0.554 0.568 0.581 0.594 0.606 0.617 0.627 0.637 0.646 0.655 0.663 0.671 0.678 0.685 0.692 0.698 0.705 0.710 0.716 0.721 0.726 0.830 0.895 0.918 0.010 0.037 0.074 0.115 0.157 0.197 0.235 0.271 0.305 0.336 0.365 0.392 0.418 0.441 0.464 0.484 0.515 0.522 0.539 0.554 0.569 0.584 0.597 0.609 0.621 0.632 0.643 0.653 0.662 0.671 0.680 0.688 0.695 0.703 0.710 0.716 0.723 0.719 0.735 0.740 0.858 0.931 0.956 5.10 Settlement Based on the Theory of Elasticity 249 Table 5.8 (Continued) mⴕ nⴕ 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00 20.00 50.00 100.00 4.5 5.0 6.0 0.010 0.036 0.073 0.114 0.155 0.195 0.233 0.269 0.302 0.333 0.362 0.389 0.415 0.438 0.461 0.482 0.516 0.520 0.537 0.554 0.569 0.584 0.597 0.611 0.623 0.635 0.646 0.656 0.666 0.676 0.685 0.694 0.702 0.710 0.717 0.725 0.731 0.738 0.744 0.750 0.878 0.962 0.990 0.010 0.036 0.073 0.113 0.154 0.194 0.232 0.267 0.300 0.331 0.359 0.386 0.412 0.435 0.458 0.479 0.496 0.517 0.535 0.552 0.568 0.583 0.597 0.610 0.623 0.635 0.647 0.658 0.669 0.679 0.688 0.697 0.706 0.714 0.722 0.730 0.737 0.744 0.751 0.758 0.896 0.989 1.020 0.010 0.036 0.072 0.112 0.153 0.192 0.229 0.264 0.296 0.327 0.355 0.382 0.407 0.430 0.453 0.474 0.484 0.513 0.530 0.548 0.564 0.579 0.594 0.608 0.621 0.634 0.646 0.658 0.669 0.680 0.690 0.700 0.710 0.719 0.727 0.736 0.744 0.752 0.759 0.766 0.925 1.034 1.072 7.0 8.0 9.0 10.0 25.0 50.0 100.0 0.010 0.036 0.072 0.112 0.152 0.191 0.228 0.262 0.294 0.324 0.352 0.378 0.403 0.427 0.449 0.470 0.473 0.508 0.526 0.543 0.560 0.575 0.590 0.604 0.618 0.631 0.644 0.656 0.668 0.679 0.689 0.700 0.710 0.719 0.728 0.737 0.746 0.754 0.762 0.770 0.945 1.070 1.114 0.010 0.036 0.072 0.112 0.152 0.190 0.227 0.261 0.293 0.322 0.350 0.376 0.401 0.424 0.446 0.466 0.471 0.505 0.523 0.540 0.556 0.571 0.586 0.601 0.615 0.628 0.641 0.653 0.665 0.676 0.687 0.698 0.708 0.718 0.727 0.736 0.745 0.754 0.762 0.770 0.959 1.100 1.150 0.010 0.036 0.072 0.111 0.151 0.190 0.226 0.260 0.291 0.321 0.348 0.374 0.399 0.421 0.443 0.464 0.471 0.502 0.519 0.536 0.553 0.568 0.583 0.598 0.611 0.625 0.637 0.650 0.662 0.673 0.684 0.695 0.705 0.715 0.725 0.735 0.744 0.753 0.761 0.770 0.969 1.125 1.182 0.010 0.036 0.071 0.111 0.151 0.189 0.225 0.259 0.291 0.320 0.347 0.373 0.397 0.420 0.441 0.462 0.470 0.499 0.517 0.534 0.550 0.585 0.580 0.595 0.608 0.622 0.634 0.647 0.659 0.670 0.681 0.692 0.703 0.713 0.723 0.732 0.742 0.751 0.759 0.768 0.977 1.146 1.209 0.010 0.036 0.071 0.110 0.150 0.188 0.223 0.257 0.287 0.316 0.343 0.368 0.391 0.413 0.433 0.453 0.468 0.489 0.506 0.522 0.537 0.551 0.565 0.579 0.592 0.605 0.617 0.628 0.640 0.651 0.661 0.672 0.682 0.692 0.701 0.710 0.719 0.728 0.737 0.745 0.982 1.265 1.408 0.010 0.036 0.071 0.110 0.150 0.188 0.223 0.256 0.287 0.315 0.342 0.367 0.390 0.412 0.432 0.451 0.462 0.487 0.504 0.519 0.534 0.549 0.583 0.576 0.589 0.601 0.613 0.624 0.635 0.646 0.656 0.666 0.676 0.686 0.695 0.704 0.713 0.721 0.729 0.738 0.965 1.279 1.489 0.010 0.036 0.071 0.110 0.150 0.188 0.223 0.256 0.287 0.315 0.342 0.367 0.390 0.411 0.432 0.451 0.460 0.487 0.503 0.519 0.534 0.548 0.562 0.575 0.588 0.600 0.612 0.623 0.634 0.645 0.655 0.665 0.675 0.684 0.693 0.702 0.711 0.719 0.727 0.735 0.957 1.261 1.499 250 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Table 5.9 Variation of F2 with m9 and n9 mⴕ nⴕ 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00 20.00 50.00 100.00 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0 0.049 0.074 0.083 0.083 0.080 0.075 0.069 0.064 0.059 0.055 0.051 0.048 0.045 0.042 0.040 0.037 0.036 0.034 0.032 0.031 0.029 0.028 0.027 0.026 0.025 0.024 0.023 0.022 0.022 0.021 0.020 0.020 0.019 0.018 0.018 0.017 0.017 0.017 0.016 0.016 0.008 0.003 0.002 0.050 0.077 0.089 0.091 0.089 0.084 0.079 0.074 0.069 0.064 0.060 0.056 0.053 0.050 0.047 0.044 0.042 0.040 0.038 0.036 0.035 0.033 0.032 0.031 0.030 0.029 0.028 0.027 0.026 0.025 0.024 0.023 0.023 0.022 0.021 0.021 0.020 0.020 0.019 0.019 0.010 0.004 0.002 0.051 0.080 0.093 0.098 0.096 0.093 0.088 0.083 0.077 0.073 0.068 0.064 0.060 0.057 0.054 0.051 0.049 0.046 0.044 0.042 0.040 0.039 0.037 0.036 0.034 0.033 0.032 0.031 0.030 0.029 0.028 0.027 0.026 0.026 0.025 0.024 0.024 0.023 0.023 0.022 0.011 0.004 0.002 0.051 0.081 0.097 0.102 0.102 0.099 0.095 0.090 0.085 0.080 0.076 0.071 0.067 0.064 0.060 0.057 0.055 0.052 0.050 0.048 0.046 0.044 0.042 0.040 0.039 0.038 0.036 0.035 0.034 0.033 0.032 0.031 0.030 0.029 0.028 0.028 0.027 0.026 0.026 0.025 0.013 0.005 0.003 0.051 0.083 0.099 0.106 0.107 0.105 0.101 0.097 0.092 0.087 0.082 0.078 0.074 0.070 0.067 0.063 0.061 0.058 0.055 0.053 0.051 0.049 0.047 0.045 0.044 0.042 0.041 0.039 0.038 0.037 0.036 0.035 0.034 0.033 0.032 0.031 0.030 0.029 0.029 0.028 0.014 0.006 0.003 0.052 0.084 0.101 0.109 0.111 0.110 0.107 0.102 0.098 0.093 0.089 0.084 0.080 0.076 0.073 0.069 0.066 0.063 0.061 0.058 0.056 0.054 0.052 0.050 0.048 0.046 0.045 0.043 0.042 0.041 0.039 0.038 0.037 0.036 0.035 0.034 0.033 0.033 0.032 0.031 0.016 0.006 0.003 0.052 0.086 0.104 0.114 0.118 0.118 0.117 0.114 0.110 0.106 0.102 0.097 0.093 0.089 0.086 0.082 0.079 0.076 0.073 0.070 0.067 0.065 0.063 0.060 0.058 0.056 0.055 0.053 0.051 0.050 0.048 0.047 0.046 0.045 0.043 0.042 0.041 0.040 0.039 0.038 0.020 0.008 0.004 0.052 0.086 0.106 0.117 0.122 0.124 0.123 0.121 0.119 0.115 0.111 0.108 0.104 0.100 0.096 0.093 0.090 0.086 0.083 0.080 0.078 0.075 0.073 0.070 0.068 0.066 0.064 0.062 0.060 0.059 0.057 0.055 0.054 0.053 0.051 0.050 0.049 0.048 0.047 0.046 0.024 0.010 0.005 3.5 4.0 0.052 0.0878 0.107 0.119 0.125 0.128 0.128 0.127 0.125 0.122 0.119 0.116 0.112 0.109 0.105 0.102 0.099 0.096 0.093 0.090 0.087 0.084 0.082 0.079 0.077 0.075 0.073 0.071 0.069 0.067 0.065 0.063 0.062 0.060 0.059 0.057 0.056 0.055 0.054 0.052 0.027 0.011 0.006 0.052 0.087 0.108 0.120 0.127 0.130 0.131 0.131 0.130 0.127 0.125 0.122 0.119 0.116 0.113 0.110 0.107 0.104 0.101 0.098 0.095 0.092 0.090 0.087 0.085 0.083 0.080 0.078 0.076 0.074 0.072 0.071 0.069 0.067 0.066 0.064 0.063 0.061 0.060 0.059 0.031 0.013 0.006 5.10 Settlement Based on the Theory of Elasticity 251 Table 5.9 (Continued) mⴕ nⴕ 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00 7.25 7.50 7.75 8.00 8.25 8.50 8.75 9.00 9.25 9.50 9.75 10.00 20.00 50.00 100.00 4.5 5.0 6.0 7.0 8.0 9.0 10.0 25.0 50.0 100.0 0.053 0.087 0.109 0.121 0.128 0.132 0.134 0.134 0.133 0.132 0.130 0.127 0.125 0.122 0.119 0.116 0.113 0.110 0.107 0.105 0.102 0.099 0.097 0.094 0.092 0.090 0.087 0.085 0.083 0.081 0.079 0.077 0.076 0.074 0.072 0.071 0.069 0.068 0.066 0.065 0.035 0.014 0.007 0.053 0.087 0.109 0.122 0.130 0.134 0.136 0.136 0.136 0.135 0.133 0.131 0.129 0.126 0.124 0.121 0.119 0.116 0.113 0.111 0.108 0.106 0.103 0.101 0.098 0.096 0.094 0.092 0.090 0.088 0.086 0.084 0.082 0.080 0.078 0.077 0.075 0.074 0.072 0.071 0.039 0.016 0.008 0.053 0.088 0.109 0.123 0.131 0.136 0.138 0.139 0.140 0.139 0.138 0.137 0.135 0.133 0.131 0.129 0.127 0.125 0.123 0.120 0.118 0.116 0.113 0.111 0.109 0.107 0.105 0.103 0.101 0.099 0.097 0.095 0.093 0.091 0.089 0.088 0.086 0.085 0.083 0.082 0.046 0.019 0.010 0.053 0.088 0.110 0.123 0.132 0.137 0.140 0.141 0.142 0.142 0.142 0.141 0.140 0.138 0.137 0.135 0.133 0.131 0.130 0.128 0.126 0.124 0.122 0.120 0.118 0.116 0.114 0.112 0.110 0.108 0.106 0.104 0.102 0.101 0.099 0.097 0.096 0.094 0.092 0.091 0.053 0.022 0.011 0.053 0.088 0.110 0.124 0.132 0.138 0.141 0.143 0.144 0.144 0.144 0.144 0.143 0.142 0.141 0.139 0.138 0.136 0.135 0.133 0.131 0.130 0.128 0.126 0.124 0.122 0.121 0.119 0.117 0.115 0.114 0.112 0.110 0.108 0.107 0.105 0.104 0.102 0.100 0.099 0.059 0.025 0.013 0.053 0.088 0.110 0.124 0.133 0.138 0.142 0.144 0.145 0.146 0.146 0.145 0.145 0.144 0.143 0.142 0.141 0.140 0.139 0.137 0.136 0.134 0.133 0.131 0.129 0.128 0.126 0.125 0.123 0.121 0.120 0.118 0.117 0.115 0.114 0.112 0.110 0.109 0.107 0.106 0.065 0.028 0.014 0.053 0.088 0.110 0.124 0.133 0.139 0.142 0.145 0.146 0.147 0.147 0.147 0.147 0.146 0.145 0.145 0.144 0.143 0.142 0.140 0.139 0.138 0.136 0.135 0.134 0.132 0.131 0.129 0.128 0.126 0.125 0.124 0.122 0.121 0.119 0.118 0.116 0.115 0.113 0.112 0.071 0.031 0.016 0.053 0.088 0.111 0.125 0.134 0.140 0.144 0.147 0.149 0.151 0.152 0.152 0.153 0.153 0.154 0.154 0.154 0.154 0.154 0.154 0.154 0.154 0.154 0.153 0.153 0.153 0.153 0.152 0.152 0.152 0.151 0.151 0.150 0.150 0.150 0.149 0.149 0.148 0.148 0.147 0.124 0.071 0.039 0.053 0.088 0.111 0.125 0.134 0.140 0.144 0.147 0.150 0.151 0.152 0.153 0.154 0.155 0.155 0.155 0.156 0.156 0.156 0.156 0.156 0.156 0.157 0.157 0.157 0.157 0.157 0.157 0.157 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.156 0.148 0.113 0.071 0.053 0.088 0.111 0.125 0.134 0.140 0.145 0.148 0.150 0.151 0.153 0.154 0.154 0.155 0.155 0.156 0.156 0.156 0.157 0.157 0.157 0.157 0.157 0.157 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.158 0.156 0.142 0.113 252 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Table 5.10 Variation of If with Df /B, B/L, and s B/L s Df/B 0.2 0.5 1.0 0.3 0.2 0.4 0.6 1.0 0.2 0.4 0.6 1.0 0.2 0.4 0.6 1.0 0.95 0.90 0.85 0.78 0.97 0.93 0.89 0.82 0.99 0.95 0.92 0.85 0.93 0.86 0.80 0.71 0.96 0.89 0.84 0.75 0.98 0.93 0.87 0.79 0.90 0.81 0.74 0.65 0.93 0.85 0.78 0.69 0.96 0.89 0.82 0.72 0.4 0.5 Due to the nonhomogeneous nature of soil deposits, the magnitude of Es may vary with depth. For that reason, Bowles (1987) recommended using a weighted average of Es in Eq. (5.33), or gEs(i) Dz (5.42) Es 5 z where Es(i) 5 soil modulus of elasticity within a depthDz z 5 H or 5B, whichever is smaller Example 5.5 Refer to Figure 5.16 and consider a rigid square foundation 2.44 m 2.44 m in plan (Df 1.22 m) on a layer of normally consolidated sand. A rock layer is located at z 10.98 m. The following is an approximation of the standard penetration number (N60) with z. z (m) N60 0–2.44 2.44–21 6.4–36 7 6.4 10.98 Given: s 0.3 and qo 167.7 kN/m2. Estimate the elastic settlement of the foundation. Use Eq. (2.29). 5.10 Settlement Based on the Theory of Elasticity 253 Solution Calculation of Average Es From Eq. (2.29), Es ⬇ pa N60 B 2.44 m, pa ⬇ 100 kN/m , and 2 Given: H 10.98 m ⬇ 10 (since it is a normally consolidated clean sand). 5B. The approximate variations of Es using Eq. (2.29) are given below. z (m) z (m) N60 Es (kN/m2) 2.44 3.96 4.58 7 11 14 7000 11,000 14,000 0–2.44 2.44–6.4 6.4–10.98 Using Eq. (5.42), Es 5 SEs(i) Dz z 5 (7,000) (2.44) 1 (11,000) (3.96) 1 (14,000) (4.58) 10.98 5 11,362 kN>m2 Calculation of Se below the Center of the Foundation [Eq. (5.33)] Se 5 qo (aBr ) 1 2 m2s II Es s f 2.44 5 1.22 m 2 a54 Br 5 L 51 B H 10.98 nr 5 5 59 B 2.44 a b a b 2 2 mr 5 Is 5 F1 1 1 2 2ms F 1 2 ms 2 From Table 5.8, F1 0.491. From Table 5.9, F2 0.017. 1 2 (2) (0.3) d (0.017) 5 0.5007 1 2 0.3 For s 3, Df /B 1.22/2.44 0.5, and B/L 1, the value of If is about 0.78 (Table 5.10). Is 5 0.491 1 c Se 5 (167.7) (4 3 1.22) a 1 2 0.32 b (0.5007) (0.78) 5 0.0256 m < 25.6 mm 11,362 254 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Calculation of Se for Rigid Foundation From Eq. (5.41), Se(rigid) ⬇ 0.93Se(flexible, center) (25.6)(0.93) 23.81 mm ⬇ 24 mm 5.11 ■ Improved Equation for Elastic Settlement In 1999, Mayne and Poulos presented an improved formula for calculating the elastic settlement of foundations. The formula takes into account the rigidity of the foundation, the depth of embedment of the foundation, the increase in the modulus of elasticity of the soil with depth, and the location of rigid layers at a limited depth. To use Mayne and Poulos’s equation, one needs to determine the equivalent diameter Be of a rectangular foundation, or Be 5 4BL Å p (5.43) where B 5 width of foundation L 5 length of foundation For circular foundations, Be 5 B (5.44) where B 5 diameter of foundation. Figure 5.17 shows a foundation with an equivalent diameter Be located at a depth Df below the ground surface. Let the thickness of the foundation be t and the modulus of elasticity of the foundation material be Ef . A rigid layer is located at a depth H below the bottom of the foundation. The modulus of elasticity of the compressible soil layer can be given as Es 5 Eo 1 kz (5.45) qo Be Df Ef t Compressible soil layer Es s Eo Es Es Eo kz H Rigid layer Depth, z Figure 5.17 Improved equation for calculating elastic settlement: general parameters 5.11 Improved Equation for Elastic Settlement 255 With the preceding parameters defined, the elastic settlement below the center of the foundation is Se 5 qoBeIGIFIE a1 2 m2s b Eo (5.46) where IG 5 influence factor for the variation of Es with depth Eo H 5 f¢b 5 , ≤ kBe Be IF 5 foundation rigidity correction factor IE 5 foundation embedment correction factor Figure 5.18 shows the variation of IG with b 5 Eo>kBe and H>Be . The foundation rigidity correction factor can be expressed as IF 5 p 1 4 1 (5.47) Ef 4.6 1 10 § Eo 1 Be k 2 ¥¢ 2t 3 ≤ Be Similarly, the embedment correction factor is IE 5 1 2 1 Be 3.5 exp(1.22ms 2 0.4) ¢ 1 1.6≤ Df (5.48) Figures 5.19 and 5.20 show the variation of IF and IE with terms expressed in Eqs. (5.47) and (5.48). 1.0 10.0 30 5.0 0.8 2.0 1.0 0.4 0.5 IG 0.6 0.2 H/Be 0.2 0 0.01 2 4 6 0.1 1 E kBo e 10 100 Figure 5.18 Variation of IG with b 256 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement 1.0 0.95 IF 0.9 0.85 KF ( Ef 3 )( ) 2t Be Be k 2 Flexibility factor 0.8 Eo 0.75 0.7 0.001 2 4 0.01 0.1 1.0 10.0 100 KF Figure 5.19 Variation of rigidity correction factor IF with flexibility factor KF [Eq. (5.47)] 1.0 0.95 IE 0.9 s = 0.5 0.4 0.85 0.3 0.2 0.8 0.1 0 0.75 0.7 0 5 10 Df Be 15 20 Figure 5.20 Variation of embedment correction factor IE with Df>Be [Eq (5.48)] Example 5.6 For a shallow foundation supported by a silty clay, as shown in Figure 5.17, Length 5 L 5 3.05 m Width 5 B 5 1.52 m Depth of foundation 5 Df 5 1.52 m 5.11 Improved Equation for Elastic Settlement 257 Thickness of foundation 5 t 5 0.305 m Load per unit area 5 qo 5 239.6 kN>m2 Ef 5 15.87 3 106 kN>m2 The silty clay soil has the following properties: H 5 3.66 m ms 5 0.3 Eo 5 9660 kN>m2 k 5 565.6 kN>m2>m Estimate the elastic settlement of the foundation. Solution From Eq. (5.43), the equivalent diameter is Be 5 (4) (1.52) (3.05) 4BL 5 5 2.43 m p p Å Å so b5 Eo 9660 5 5 7.02 kBe (565.6) (2.43) and 3.66 H 5 5 1.5 Be 2.43 From Figure 5.18, for b 5 7.02 and H>Be 5 1.5, the value of IG < 0.69. From Eq. (5.47), IF 5 p 1 4 1 4.6 1 10 § Ef Eo 1 5 Be k 2 p 1 4 ¥¢ 2t 3 ≤ Be 1 4.6 1 10 D 5 0.785 15.87 3 106 9660 1 ¢ 2.43 ≤ (565.6) 2 TB (2) (0.305) 3 R 2.43 From Eq. (5.48 ), IE 5 1 2 1 3.5 exp(1.22ms 2 0.4) ¢ Be 1 1.6≤ Df 258 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement 512 1 2.43 3.5 exp 3(1.22) (0.3) 2 0.44 ¢ 1 1.6≤ 1.52 5 0.908 From Eq. (5.46), Se 5 qoBeIGIFIE (1 2 m2s ) Eo so, with qo 5 239.6 kN>m2, it follows that Se 5 5.12 (239.6) (2.43) (0.69) (0.785) (0.908) (1 2 0.32 ) 5 0.027 m 5 27 mm (9660) ■ Settlement of Sandy Soil: Use of Strain Influence Factor The settlement of granular soils can also be evaluated by the use of a semiempirical strain influence factor proposed by Schmertmann et al. (1978). According to this method (Figure 5.21), the settlement is z2 I z Se 5 C1C2 (q 2 q) a Dz E 0 s (5.49) where Iz 5 strain influence factor C1 5 a correction factor for the depth of foundation embedment 5 1 2 0.5 3q> (q 2 q)4 C2 5 a correction factor to account for creep in soil 5 1 1 0.2 log (time in years>0.1) q 5 stress at the level of the foundation q 5 gDf 5 effective stress at the base of the foundation Es 5 modulus of elasticity of soil The recommended variation of the strain influence factor Iz for square (L> B 5 1) or circular foundations and for foundations with L> B $ 10 is shown in Figure 5.21. The Iz diagrams for 1 , L> B , 10 can be interpolated. Note that the maximum value of Iz [that is, Iz(m)] occurs at z z1 and then reduces to zero at z z2. The maximum value of Iz can be calculated as where q# 2 q Iz(m) 5 0.5 1 0.1 Ä qz(1) r q z(1) effective stress at a depth of z1 before construction of the foundation (5.50) 259 5.12 Settlement of Sandy Soil: Use of Strain Influence Factor q Df Iz (m) Iz (m) q = Df Iz 0.1 0.2 Iz qz(1) B z1 = 0.5B qz(1) z1 = B z2 = 2B L/B = 1 z z L/B ≥ 10 z2 = 4B z Figure 5.21 Variation of strain influence factor with depth and L/B The following relations are suggested by Salgado (2008) for interpolation of Iz at z 0, z1/B, and z2/B for rectangular foundations. • Iz at z 0 L 2 1b # 0.2 B (5.51) z1 L 5 0.5 1 0.0555a 2 1b # 1 B B (5.52) z2 L 5 2 1 0.222a 2 1b # 4 B B (5.53) Iz 5 0.1 1 0.0111a • • Variation of z1/B for Iz(m) Variation of z2/B Schmertmann et al. (1978) suggested that Es 2.5qc (for square foundation) (5.54) and Es 3.5qc (for L/B where qc cone penetration resistance. 10) (5.55) 260 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement It appears reasonable to write (Terzaghi et al., 1996) L Es( rectangle) 5 a1 1 0.4 log bEs( square) B (5.56) The procedure for calculating elastic settlement using Eq. (5.49) is given here (Figure 5.22). Step 1. Plot the foundation and the variation of Iz with depth to scale (Figure 5.22a). Step 2. Using the correlation from standard penetration resistance (N60) or cone penetration resistance (qc), plot the actual variation of Es with depth (Figure 5.22b). Step 3. Approximate the actual variation of Es into a number of layers of soil having a constant Es, such as Es(1), Es(2), . . . , Es(i), . . . Es(n) (Figure 5.22b). Step 4. Divide the soil layer from z 5 0 to z 5 z2 into a number of layers by drawing horizontal lines. The number of layers will depend on the break in continuity in the Iz and Es diagrams. Iz Step 5. Prepare a table (such as Table 5.11) to obtain S Dz. Es Step 6. Calculate C1 and C2. Step 7. Calculate Se from Eq. (5.49). B Df z(1) Es Es(1) Iz(1) Step 4 z1 z(2) Iz(2) Es(2) Step 3 Iz(3) z2 z(i) z(n) Iz(i) Es(i) Step 1 Iz(n) Depth, z (a) Es(n) Depth, z (b) Figure 5.22 Procedure for calculation of Se using the strain influence factor Step 2 5.12 Settlement of Sandy Soil: Use of Strain Influence Factor Table 5.11 Calculation of S Iz Es Dz Layer no. Dz Es Iz at the middle of the layer 1 2 Dz(1) Dz(2) Es(1) Es(2) Iz(1) Iz(2) ( ( ( ( i Dz(i) Es(i) Iz(i) ( ( ( ( n Dz(n) 261 Es(n) Iz(n) Iz Dz Es Iz(1) Es(1) Iz(i) Es(i) Dz1 Dzi ( Iz(n) Es(n) S Iz Es Dzn Dz Example 5.7 Consider a rectangular foundation 2 m 4 m in plan at a depth of 1.2 m in a sand deposit, as shown in Figure 5.23a. Given: 17.5 kN/m3; q– 145 kN/m2, and the following approximated variation of qc with z: z (m) qc (kN/m2) 0–0.5 0.5–2.5 2.5–5.0 2250 3430 2950 Estimate the elastic settlement of the foundation using the strain influence factor method. Solution From Eq. (5.52), z1 L 4 5 0.5 1 0.0555a 2 1b 5 0.5 1 0.0555a 2 1b < 0.56 B B 2 z1 (0.56)(2) 1.12 m From Eq. (5.53), z2 L 5 2 1 0.222a 2 1b 5 2 1 0.222(2 2 1) 5 2.22 B B z2 (2.22)(2) 4.44 m 262 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement From Eq. (5.51), at z 0, Iz 5 0.1 1 0.0111a L 4 2 1b 5 0.1 1 0.0111a 2 1b < 0.11 B 2 From Eq. (5.50), q# 2 q 145 2 (1.2 3 17.5) 0.5 Iz(m) 5 0.5 1 0.1 5 0.5 1 0.1 c d 5 0.675 Å q9z(1) (1.2 1 1.12) (17.5) The plot of Iz versus z is shown in Figure 5.23c. Again, from Eq. (5.56) L 4 Es(rectangle) 5 a1 1 0.4log bEs(square) 5 c1 1 0.4loga b d (2.5 3 qc ) 5 2.8qc B 2 Hence, the approximated variation of Es with z is as follows: z (m) qc (kN/m2) Es (kN/m2) 0–0.5 0.5–2.5 2.5–5.0 2250 3430 2950 6300 9604 8260 The plot of Es versus z is shown in Figure 5.23b. q = 145 kN/m2 1.2 m B=2 m L=4 m = 17.5 kN/m3 0.5 0.675 0.11 6300 kN/m2 1.0 2.0 z Es (kN/m2) 1 2 1.12 9604 kN/m2 3 2.5 3.0 8260 kN/m2 (a) 4 4.0 4.44 5.0 z (m) (b) Figure 5.23 z (m) (c) Iz 5.13 Settlement of Foundation on Sand Based on Standard Penetration Resistance 263 The soil layer is divided into four layers as shown in Figures 5.23b and 5.23c. Now the following table can be prepared. Layer no. z (m) Es (kN/m2) Iz at middle of layer 1 2 3 4 0.50 0.62 1.38 1.94 6300 9604 9604 8260 0.236 0.519 0.535 0.197 Se 5 C1C2 (q 2 q) a C1 5 1 2 0.5a Iz Es Iz Dz (m3> kN) Es 1.87 105 3.35 105 7.68 105 4.62 105 17.52 105 Dz q 21 b 5 1 2 0.5a b 5 0.915 q2q 145 2 21 Assume the time for creep is 10 years. So, C2 5 1 1 0.2loga 10 b 5 1.4 0.1 Hence, Se (0.915)(1.4)(145 21)(17.52 105) 2783 105 m 27.83 mm 5.13 ■ Settlement of Foundation on Sand Based on Standard Penetration Resistance Meyerhof’s Method Meyerhof (1956) proposed a correlation for the net bearing pressure for foundations with the standard penetration resistance, N60. The net pressure has been defined as qnet 5 q 2 gDf where q 5 stress at the level of the foundation. According to Meyerhof’s theory, for 25 mm (1 in.) of estimated maximum settlement, N60 (for B # 1.22 m) 0.08 (5.57) N60 B 1 0.3 2 ¢ ≤ (for B . 1.22 m) 0.125 B (5.58) qnet (kN>m2 ) 5 and qnet (kN>m2 ) 5 264 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Since the time that Meyerhof proposed his original correlations, researchers have observed that its results are rather conservative. Later, Meyerhof (1965) suggested that the net allowable bearing pressure should be increased by about 50%. Bowles (1977) proposed that the modified form of the bearing equations be expressed as qnet (kN>m2 ) 5 N60 Se Fa b 2.5 d 25 (for B # 1.22 m) (5.59) and qnet (kN>m2 ) 5 N60 B 1 0.3 2 Se ¢ ≤ Fd a b (for B . 1.22 m) 0.08 B 25 (5.60) where Fd 5 depth factor 5 1 1 0.33(Df > B) B 5 foundation width, in meters Se 5 settlement, in mm Hence, Se (mm) 5 1.25qnet (kN>m2 ) N60Fd (for B < 1.22 m) (5.61) and Se (mm) 5 2 2qnet (kN>m2 ) B ¢ ≤ N60Fd B 1 0.3 (for B . 1.22 m) (5.62) The N60 referred to in the preceding equations is the standard penetration resistance between the bottom of the foundation and 2B below the bottom. Burland and Burbidge’s Method Burland and Burbidge (1985) proposed a method of calculating the elastic settlement of sandy soil using the field standard penetration number, N60 . (See Chapter 2.) The method can be summarized as follows: 1. Variation of Standard Penetration Number with Depth Obtain the field penetration numbers (N60 ) with depth at the location of the foundation. The following adjustments of N60 may be necessary, depending on the field conditions: For gravel or sandy gravel, N60(a) < 1.25 N60 (5.63) For fine sand or silty sand below the groundwater table and N60 . 15, N60(a) < 15 1 0.5(N60 2 15) (5.64) where N60(a) 5 adjusted N60 value. 2. Determination of Depth of Stress Influence (zzr) In determining the depth of stress influence, the following three cases may arise: 265 5.13 Settlement of Foundation on Sand Based on Standard Penetration Resistance Case I. If N60 [or N60(a)] is approximately constant with depth, calculate zr from zr B 0.75 5 1.4¢ ≤ BR BR (5.65) where BR 5 reference width 5 0.3 m (if B is in m) B 5 width of the actual foundation Case II. If N60 [or N60(a)] is increasing with depth, use Eq. (5.65) to calculate zr. Case III. If N60 [or N60(a)] is decreasing with depth, zr 5 2B or to the bottom of soft soil layer measured from the bottom of the foundation (whichever is smaller). 3. Calculation of Elastic Settlement Se The elastic settlement of the foundation, Se , can be calculated from 2 Se 5 a1a2a3 E BR L 1.25¢ ≤ B L 0.25 1 ¢ ≤ B U ¢ B 0.7 qr ≤ ¢ ≤ pa BR (5.66) where a1 5 a constant a2 5 compressibility index a3 5 correction for the depth of influence pa 5 atmospheric pressure 5 100 kN>m2 L 5 length of the foundation Table 5.12 summarizes the values of q , 1, 2, and 3 to be used in Eq. (5.70) for 2 2 various types of soils. Note that, in this table, N 60 or N 60(a) average value of N60 or N60(a) in the depth of stress influence. Table 5.12 Summary of 1, 2, and 3 Soil type q ␣1 ␣2 Normally consolidated sand qnet 0.14 1.71 2 or N 2 4 1.4 3N 60 60(a) a3 5 Overconsolidated sand (qnet c ) qnet 0.047 0.57 2 or N 2 4 1.4 3N 60 60(a) or H H a2 2 9 b zr z (if H z ) 3 1 (if H z) where H depth of compressible layer where c preconsolidation pressure Overconsolidated sand (qnet c ) ␣3 qnet 0.67c 0.14 0.57 2 or N 2 4 1.4 3N 60 60(a) 266 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Example 5.8 A shallow foundation measuring 1.75 m 3 1.75 m is to be constructed over a layer of sand. Given Df 5 1 m; N60 is generally increasing with depth; N60 in the depth of stress influence 5 10, qnet 5 120 kN> m2. The sand is normally consolidated. Estimate the elastic settlement of the foundation. Use the Burland and Burbidge method. Solution From Eq. (5.69), zr B 0.75 5 1.4a b BR BR Depth of stress influence, zr 5 1.4a B 0.75 1.75 0.75 b BR 5 (1.4) (0.3) a b < 1.58 m BR 0.3 From Eq. (5.70), 2 L 1.25a b Se B B 0.7 qr 5 a1a2a3 D T a b a b pa BR L BR 0.25 1 a b B For normally consolidated sand (Table 5.12), a1 5 0.14 1.71 1.71 a2 5 5 5 0.068 1.4 (N60 ) (10) 1.4 a3 5 1 qr 5 qnet 5 120 kN>m2 So, (1.25) a 2 1.75 b Se 1.75 1.75 0.7 120 5 (0.14) (0.068) (1) D b a b T a 0.3 0.3 100 1.75 0.25 1 a b 1.75 Se < 0.0118 m 5 11.8 mm ■ 5.14 Settlement in Granular Soil Based on Pressuremeter Test (PMT) 267 Example 5.9 Solve Example 5.8 using Meyerhof ’s method. Solution From Eq. (5.66), 2 2qnet B ¢ ≤ (N60 ) (Fd ) B 1 0.3 Fd 5 1 1 0.33(Df>B) 5 1 1 0.33(1>1.75) 5 1.19 Se 5 Se 5 5.14 2 (2) (120) 1.75 ¢ ≤ 5 14.7 mm (10) (1.19) 1.75 1 0.3 ■ Settlement in Granular Soil Based on Pressuremeter Test (PMT) Briaud (2007) proposed a method based on Pressuremeter tests (Section 2.22) from which the load-settlement diagrams of foundations can be derived. The following is a step-by-step procedure for performing the analysis. Step 1. Step 2. Step 3. Conduct Pressuremeter tests at varying depths at the desired location and obtain plots of pp (pressure in the measuring cell for cavity expansion; see Figure 2.32) versus R/Ro (Ro initial radius of the PMT cavity, and R increase in the cavity radius), as shown in Figure 5.24a. Extend the straight line part of the PMT curve to zero pressure and shift the vertical axis, as shown in Figure 5.24a. Re-zero the R/Ro axis. Draw a strain influence factor diagram for the desired foundation (Section 5.12). Using all Pressuremeter test curves within the depth of influence, develop a mean PMT curve. Referring to Figure 5.24b, this can be done as follows: For each value of R/Ro, let the pp values be pp(1), pp(2), pp(3), . . . . The mean value of pp can be obtained as pp(m) 5 A1 A2 A3 pp(1) 1 pp(2) 1 p 1 c# A A A p(3) (5.67) where A1, A2, A3 areas tributary to each test under the strain influence factor diagram A A1 A2 A3 c# (5.68) Step 4. Based on the results of Step 3, develop a mean pp(m) versus R/Ro plot (Figure 5.24c). 268 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement pp (a) New origin for R R Ro Ro Strain influence factor, Iz A1 A2 (b) A3 PMT 1 2 3 Depth, z pp(m) (c) R Ro Figure 5.24 (a) Plot of pp versus R/Ro; (b) averaging the pressuremeter curves within the foundation zone of influence; (c) plot of pp(m) versus R/Ro Step 5. The mean PMT curve now can be used to develop the load-settlement plot for the foundation via the following equations. Se DR 5 0.24 B Ro (5.69) qo 5 fL>Bfefdfb,dGpp(m) (5.70) and where Se 5 elastic settlement of the foundation B 5 width of foundation L 5 length of foundation qo 5 net load per unit area on the foundation G 5 gamma function linking qo and pp(m) 269 5.14 Settlement in Granular Soil Based on Pressuremeter Test (PMT) B fL>B 5 shape factor 5 0.8 1 0.2a b L (5.71) e fe 5 eccentricity factor 5 1 2 0.33a b (center) B 0.5 e fe 5 eccentricity factor 5 1 2 a b (edge) B d(deg) 2 fd 5 load inclination factor 5 1 2 c d (center) 90 d(deg) 0.5 fd 5 load inclination factor 5 1 2 c d (edge) 360 d 0.1 fb,d 5 slope factor 5 0.8a1 1 b (3H:1V slope) B d 0.15 fb,d 5 slope factor 5 0.7a1 1 b ( 2H:1V slope) B (5.72) (5.73) (5.74) (5.75) (5.76) (5.77) inclination of load with respect to the vertical inclination of a slope with the horizontal if the foundation is located on top of a slope d distance of the edge of the foundation from the edge of the slope The parameters , , d, and e are defined in Figure 5.25. Figure 5.26 shows the design plot DR for with Se/B or 0.24 . Ro Γ 0 0 1 2 0.02 ΔR 4.2 Ro or Se B 0.04 0.06 Q e δ β d 0.08 Foundation B×L B Figure 5.25 Definition of parameters—, L, d, , , e, and 0.1 Figure 5.26 Variation of with Se/B 0.24 R/Ro 3 270 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Step 6. Based on the values of B/L, e/B, , and d/B, calculate the values of fL/B, fe, f, and f,d as needed. Let f 5 (fL>B ) (fe ) (fd ) (fb,d ) (5.78) qo 5 fGpp(m) (5.79) Thus, Step 7. Step 8. Now prepare a table, as shown in Table 5.13. Complete Table 5.13 as follows: a. Column 1—Assume several values of R/Ro. b. Column 2—For given values of R/Ro, obtain pp(m) from Figure 5.24c. c. Column 3—From Eq. (5.73), calculate the values of Se/B from values of R/Ro given in Column 1. d. Column 4—With known values of B, calculate the values of Se. e. Column 5—From Figure 5.26, obtain the desired values of . f. Column 6—Use Eq. (5.83) to obtain qo. g. Now plot a graph of Se (Column 4) versus qo (Column 6) from which the magnitude of Se for a given qo can be determined. Table 5.13 Calculations to Obtain the Load-Settlement Plot R/Ro (1) pp(m) (2) Se/B (3) Se (4) (5) qo (6) Example 5.10 A spread footing, shown in Figure 5.27a with a width of 4 m and a length of 20 m, serves as a bridge abutment foundation. The soil is medium dense sand. A 16,000 kN vertical load acts on the footing. The active pressure on the abutment wall develops a 1,600 kN horizontal load. The resultant reaction force due to the vertical and horizontal load is applied at an eccentricity of 0.13 m. PMT testing at the site produced a mean Pressuremeter curve characterizing the soil and is shown in Figure 5.27b. What is the settlement at the current loading? 5.14 Settlement in Granular Soil Based on Pressuremeter Test (PMT) V = 16,000 kN 4m 1 H = 1600 kN 3 d=3 m L = 20 m e = 0.13 m B=4 m (a) 1800 1600 pp (m) (kN/m2) 1400 pp (m) 1200 1000 800 600 400 200 ΔR/Ro (kN/m2) 0.002 0.005 0.01 0.02 0.04 0.07 0.1 0.2 50 150 250 450 800 1200 1400 1700 0 0 0.05 0.1 ΔR/Ro (b) 0.15 0.2 Figure 5.27 Solution Given: B 4 m, L 20 m, d 3 m, and slope 3H:1V. So 4 B fL>B 5 0.8 1 0.2a b 5 0.8 1 0.2a b 5 0.84 L 20 e 0.13 fe(center) 5 1 2 0.33a b 5 1 2 0.33a b 5 0.99 B 4 fd(center) 5 1 2 a d 2 b 90 271 272 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement d 5 tan21 a fd 5 1 2 a H 1600 b 5 tan21 a b 5 5.71° V 16,000 5.71 2 b 5 0.996 90 fb,d 5 0.8a1 1 d 0.1 3 0.1 b 5 0.8a1 1 b 5 0.846 B 4 f fL/Bfe f f,d (0.84)(0.99)(0.996)(0.845) 0.7 Now the following table can be prepared. R /Ro (1) pp(m) (kN/m2) (2) Se/B (3) Se (mm) (4) (5) qo (kN/m2) (6) Qo (MN) (7) 0.002 0.005 0.01 0.02 0.04 0.07 0.10 0.20 50 150 250 450 800 1200 1400 1700 0.0005 0.0012 0.0024 0.0048 0.0096 0.0168 0.024 0.048 2.0 4.8 9.6 19.2 38.4 67.2 96.0 192.0 2.27 2.17 2.07 1.83 1.40 1.17 1.07 0.90 79.45 227.85 362.25 576.45 784.00 982.8 1048.6 1071.0 6.36 18.23 28.98 46.12 62.72 78.62 83.89 85.68 Note: Columns 1 and 2: From Figure 5.27b Column 3: (Column 1)(0.24) Se/B Column 4: (Column 3)(B 4000 mm) Se Column 5: From Figure 5.26 Column 6: f pp(m) (0.7)()pp(m) qo Column 7: (Column 6)(B L) Qo Figure 5.28 shows the plot of Qo versus Se. From this plot it can be seen that, for a vertical loading of 16,000 kN (16 MN), the value of Se ⬇ 4.2 mm. Qo (MN) 0 0 20 40 60 80 100 20 40 Se (mm) 60 80 100 120 140 160 180 200 Figure 5.28 ■ 68125_05_ch5_p223-290.qxd 3/12/10 11:16 AM Page 273 5.15 Primary Consolidation Settlement Relationships 273 Consolidation Settlement 5.15 Primary Consolidation Settlement Relationships As mentioned before, consolidation settlement occurs over time in saturated clayey soils subjected to an increased load caused by construction of the foundation. (See Figure 5.29.) On the basis of the one-dimensional consolidation settlement equations given in Chapter 1, we write Sc(p) 5 3ezdz (5.80) where ez 5 vertical strain De 5 1 1 eo De 5 change of void ratio 5 f(sor , scr , and Dsr) So, Sc(p) 5 CcHc sor 1 Dsav r log 1 1 eo sor (for normally consolidated clays) qo Stress increase, Groundwater table t Clay layer Hc m b Depth, z Figure 5.29 Consolidation settlement calculation (5.81) 274 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Sc(p) 5 CsHc sor 1 Dsav r log 1 1 eo sor (for overconsolidated clays with sor 1 Dsav r , scr ) (5.82) Sc(p) 5 CsHc scr CcHc sor 1 Dsav r log 1 log 1 1 eo sor 1 1 eo scr (for overconsolidated clays with sor , scr , sor 1 Dsav r) (5.83) where sor 5 average effective pressure on the clay layer before the construction of the foundation Dsav r 5 average increase in effective pressure on the clay layer caused by the construction of the foundation scr 5 preconsolidation pressure eo 5 initial void ratio of the clay layer Cc 5 compression index Cs 5 swelling index Hc 5 thickness of the clay layer The procedures for determining the compression and swelling indexes were discussed in Chapter 1. Note that the increase in effective pressure, Dsr, on the clay layer is not constant with depth: The magnitude of Dsr will decrease with the increase in depth measured from the bottom of the foundation. However, the average increase in pressure may be approximated by Dsav r 5 16 ( Dstr 1 4Dsm r 1 Dsbr ) (5.84) where Dstr , Dsm r , and Dsbr are, respectively, the effective pressure increases at the top, middle, and bottom of the clay layer that are caused by the construction of the foundation. The method of determining the pressure increase caused by various types of foundation load using Boussinesq’s solution is discussed in Sections 5.2 through 5.6. Dsav r can also be directly obtained from the method presented in Section 5.5. 5.16 Three-Dimensional Effect on Primary Consolidation Settlement The consolidation settlement calculation presented in the preceding section is based on Eqs. (1.61), (1.63), and (1.65). These equations, as shown in Chapter 1, are in turn based on one-dimensional laboratory consolidation tests. The underlying assumption is that the increase in pore water pressure, Du, immediately after application of the load equals the increase in stress, Ds, at any depth. In this case, De Sc(p) 2oed 5 3 dz 5 3mv Ds(1) r dz 1 1 eo where Sc(p) 2oed 5 consolidation settlement calculated by using Eqs. (1.61), (1.63), and (1.65) Ds(1) r 5 effective vertical stress increase mv 5 volume coefficient of compressibility (see Chapter 1) 5.16 Three-Dimensional Effect on Primary Consolidation Settlement 275 Flexible circular load Clay (1) Hc (3) z (3) Figure 5.30 Circular foundation on a clay layer In the field, however, when a load is applied over a limited area on the ground surface, such an assumption will not be correct. Consider the case of a circular foundation on a clay layer, as shown in Figure 5.30. The vertical and the horizontal stress increases at a point in the layer immediately below the center of the foundation are Ds(1) and Ds(3) , respectively. For a saturated clay, the pore water pressure increase at that depth (see Chapter 1) is Du 5 Ds(3) 1 A3Ds(1) 2 Ds(3) 4 (5.85) Sc(p) 5 3mv Du dz 5 3 (mv ) 5Ds(3) 1 A3Ds(1) 2 Ds(3) 46 dz (5.86) where A 5 pore water pressure parameter. For this case, Thus, we can write Hc Kcir 5 Sc(p) Sc(p) 2oed Hc 3 mv Du dz 5 0 Hc r dz 3 Ds(3) 5 A 1 (1 2 A) D r dz 3 mv Ds(1) 0 Hc T (5.87) r dz 3 Ds(1) 0 0 where Kcir 5 settlement ratio for circular foundations. The settlement ratio for a continuous foundation, Kstr , can be determined in a manner similar to that for a circular foundation. The variation of Kcir and Kstr with A and Hc>B is given in Figure 5.31. (Note: B 5 diameter of a circular foundation, and B 5 width of a continuous foundation.) The preceding technique is generally referred to as the Skempton–Bjerrum modification (1957) for a consolidation settlement calculation. Leonards (1976) examined the correction factor Kcr for a three-dimensional consolidation effect in the field for a circular foundation located over overconsolidated clay. Referring to Figure 5.30, we have Sc(p) 5 Kcr(OC) Sc(p) 2oed (5.88) B ≤ Hc (5.89) where Kcr(OC) 5 f¢OCR, 276 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement 1.0 H c/B = Settlement ratio 0.8 0.25 0.25 0.5 0.5 0.6 1.0 2.0 1.0 0.4 2.0 Circular foundation 0.2 Continuous foundation 0 0 0.2 0.4 0.6 0.8 Pore water pressure parameter, A 1.0 Figure 5.31 Settlement ratios for circular (Kcir ) and continuous (Kstr ) foundations in which OCR 5 overconsolidation ratio 5 scr sor (5.90) where scr 5 preconsolidation pressure sor 5 present average effective pressure The interpolated values of Kcr(OC) from Leonard’s 1976 work are given in Table 5.14. Table 5.14 Variation of Kcr(OC) with OCR and B>Hc Kcr (OC ) OCR B/Hc ⴝ 4.0 B/Hc ⴝ 1.0 B/Hc ⴝ 0.2 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 0.986 0.972 0.964 0.950 0.943 0.929 0.914 0.900 0.886 0.871 0.864 0.857 0.850 0.843 0.843 1 0.957 0.914 0.871 0.829 0.800 0.757 0.729 0.700 0.671 0.643 0.629 0.614 0.607 0.600 0.600 1 0.929 0.842 0.771 0.707 0.643 0.586 0.529 0.493 0.457 0.429 0.414 0.400 0.386 0.371 0.357 5.16 Three-Dimensional Effect on Primary Consolidation Settlement 277 Example 5.11 A plan of a foundation 1 m 3 2 m is shown in Figure 5.32. Estimate the consolidation settlement of the foundation, taking into account the three-dimensional effect. Given: A 5 0.6. Solution The clay is normally consolidated. Thus, Sc(p)2oed 5 CcHc sor 1 Dsav r log 1 1 eo sor so sor 5 (2.5) (16.5) 1 (0.5) (17.5 2 9.81) 1 (1.25) (16 2 9.81) 5 41.25 1 3.85 1 7.74 5 52.84 kN>m2 From Eq. (5.84), Dsav r 5 16 (Dstr 1 4Dsm r 1 Dsbr ) Now the following table can be prepared (Note: L 5 2 m; B 5 1 m): a b m1 ⴝ L/B z(m) z/(B/2) ⴝ n1 Ica ⌬s9 ⴝ qoIcb 2 2 2 2 2 1 2.5>2 5 3.25 2 1 2.5 5 4.5 4 6.5 9 0.190 < 0.085 0.045 28.5 5 Dstr r 12.75 5 Dsm 6.75 5 Dsbr Table 5.3 Eq. (5.10) q0 150 kN/m2 (net stress increase) 1m BLlm2m 1.5 m Sand 16.5 kN/m3 Groundwater table 0.5 m 2.5 m Sand sat 17.5 kN/m3 Normally consolidated clay ea 0.8 16 kN/m3 s 6,000 kN/m2 Ce 0.32 s 0.5 Cs 0.09 Figure 5.32 Calculation of primary consolidation settlement for a foundation 278 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Now, Dsav r 5 16 (28.5 1 4 3 12.75 1 6.75) 5 14.38 kN>m2 so Sc(p)2oed 5 (0.32) (2.5) 52.84 1 14.38 log ¢ ≤ 5 0.0465 m 1 1 0.8 52.84 5 46.5 mm Now assuming that the 2:1 method of stress increase (see Figure 5.5) holds good, the area of distribution of stress at the top of the clay layer will have dimensions B9 5 width 5 B 1 z 5 1 1 (1.5 1 0.5) 5 3 m and L9 5 width 5 L 1 z 5 2 1 (1.5 1 0.5) 5 4 m The diameter of an equivalent circular area, Beq, can be given as p 2 Beq 5 BrLr 4 so that Beq 5 Å 4BrLr (4) (3) (4) 5 5 3.91 m p p Å Also, Hc 2.5 5 5 0.64 Beq 3.91 From Figure 5.31, for A 5 0.6 and Hc>Beq 5 0.64, the magnitude of Kcr < 0.78. Hence, Se(p) 5 KcrSe(p) – oed 5 (0.78)(46.5) < 36.3 mm 5.17 ■ Settlement Due to Secondary Consolidation At the end of primary consolidation (i.e., after the complete dissipation of excess pore water pressure) some settlement is observed that is due to the plastic adjustment of soil fabrics. This stage of consolidation is called secondary consolidation. A plot of deformation against the logarithm of time during secondary consolidation is practically linear Void ratio, e 5.17 Settlement Due to Secondary Consolidation C 279 e t log t2 1 ep e t1 Time, t (log scale) t2 Figure 5.33 Variation of e with log t under a given load increment, and definition of secondary compression index as shown in Figure 5.33. From the figure, the secondary compression index can be defined as Ca 5 De De 5 log t2 2 log t1 log (t2>t1 ) (5.91) where Ca 5 secondary compression index De 5 change of void ratio t1 , t2 5 time The magnitude of the secondary consolidation can be calculated as Sc(s) 5 C ar Hc log(t2>t1 ) (5.92) where C ar 5 Ca> (1 1 ep ) ep 5 void ratio at the end of primary consolidation Hc 5 thickness of clay layer (5.93) Mesri (1973) correlated C ar with the natural moisture content (w) of several soils, from which it appears that C ar < 0.0001w (5.94) where w 5 natural moisture content, in percent. For most overconsolidated soils, C ar varies between 0.0005 to 0.001. Mesri and Godlewski (1977) compiled the magnitude of Ca> Cc (Cc 5 compression index) for a number of soils. Based on their compilation, it can be summarized that • For inorganic clays and silts: Ca>Cc < 0.04 6 0.01 280 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement • For organic clays and silts: Ca>Cc < 0.05 6 0.01 • For peats: Ca>Cc < 0.075 6 0.01 Secondary consolidation settlement is more important in the case of all organic and highly compressible inorganic soils. In overconsolidated inorganic clays, the secondary compression index is very small and of less practical significance. There are several factors that might affect the magnitude of secondary consolidation, some of which are not yet very clearly understood (Mesri, 1973). The ratio of secondary to primary compression for a given thickness of soil layer is dependent on the ratio of the stress increment, Dsr, to the initial effective overburden stress, sor . For small Dsr>sor ratios, the secondary-to-primary compression ratio is larger. 5.18 Field Load Test The ultimate load-bearing capacity of a foundation, as well as the allowable bearing capacity based on tolerable settlement considerations, can be effectively determined from the field load test, generally referred to as the plate load test. The plates that are used for tests in the field are usually made of steel and are 25 mm thick and 150 mm to 762 mm in diameter. Occasionally, square plates that are 305 mm 3 305 mm are also used. To conduct a plate load test, a hole is excavated with a minimum diameter of 4B (B is the diameter of the test plate) to a depth of Df , the depth of the proposed foundation. The plate is placed at the center of the hole, and a load that is about one-fourth to one-fifth of the estimated ultimate load is applied to the plate in steps by means of a jack. A schematic diagram of the test arrangement is shown in Figure 5.34a. During each step of the application of the load, the settlement of the plate is observed on dial gauges. At least one hour is allowed to elapse between each application. The test should be conducted until failure, or at least until the plate has gone through 25 mm (1 in.) of settlement. Figure 5.34b shows the nature of the load–settlement curve obtained from such tests, from which the ultimate load per unit area can be determined. Figure 5.35 shows a plate load test conducted in the field. For tests in clay, qu(F) 5 qu(P) (5.95) where qu(F) 5 ultimate bearing capacity of the proposed foundation qu(P) 5 ultimate bearing capacity of the test plate Equation (5.95) implies that the ultimate bearing capacity in clay is virtually independent of the size of the plate. 5.18 Field Load Test 281 Reaction beam Jack Dial gauge Test plate diameter B Anchor pile At least 4B (a) Load/unit area Settlement Figure 5.34 Plate load test: (a) test arrangement; (b) nature of load– settlement curve (b) For tests in sandy soils, qu(F) 5 qu(P) BF BP (5.96) where BF 5 width of the foundation BP 5 width of the test plate The allowable bearing capacity of a foundation, based on settlement considerations and for a given intensity of load, qo , is SF 5 SP BF BP (for clayey soil) (5.97) and SF 5 SP ¢ 2 2BF ≤ BF 1 BP (for sandy soil) (5.98) 282 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Figure 5.35 Plate load test in the field (Courtesy of Braja M. Das, Henderson, NV) 5.19 Presumptive Bearing Capacity Several building codes (e.g., the Uniform Building Code, Chicago Building Code, and New York City Building Code) specify the allowable bearing capacity of foundations on various types of soil. For minor construction, they often provide fairly acceptable guidelines. However, these bearing capacity values are based primarily on the visual classification of near-surface soils and generally do not take into consideration factors such as the stress history of the soil, the location of the water table, the depth of the foundation, and the tolerable settlement. So, for large construction projects, the codes’ presumptive values should be used only as guides. 5.20 Tolerable Settlement of Buildings In most instances of construction, the subsoil is not homogeneous and the load carried by various shallow foundations of a given structure can vary widely. As a result, it is reasonable to expect varying degrees of settlement in different parts of a given building. 5.20 Tolerable Settlement of Buildings 283 L lAB B C D A A E E max ST (max) B D ST (max) C max Figure 5.36 Definition of parameters for differential settlement The differential settlement of the parts of a building can lead to damage of the superstructure. Hence, it is important to define certain parameters that quantify differential settlement and to develop limiting values for those parameters in order that the resulting structures be safe. Burland and Worth (1970) summarized the important parameters relating to differential settlement. Figure 5.36 shows a structure in which various foundations, at A, B, C, D, and E, have gone through some settlement. The settlement at A is AAr, at B is BBr, etc. Based on this figure, the definitions of the various parameters are as follows: ST 5 total settlement of a given point DST 5 difference in total settlement between any two points a 5 gradient between two successive points DST(ij) b 5 angular distortion 5 lij (Note: lij 5 distance between points i and j) v 5 tilt D 5 relative deflection (i.e. movement from a straight line joining two reference points) D 5 deflection ratio L Since the 1950s, various researchers and building codes have recommended allowable values for the preceding parameters. A summary of several of these recommendations is presented next. In 1956, Skempton and McDonald proposed the following limiting values for maximum settlement and maximum angular distortion, to be used for building purposes: Maximum settlement, ST(max) In sand In clay 32 mm 45 mm 284 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement Maximum differential settlement, DST(max) Isolated foundations in sand Isolated foundations in clay Raft in sand Raft in clay Maximum angular distortion, bmax 51 mm 76 mm 51–76 mm 76–127 mm 1>300 On the basis of experience, Polshin and Tokar (1957) suggested the following allowable deflection ratios for buildings as a function of L>H, the ratio of the length to the height of a building: D>L 5 0.0003 for L>H # 2 D>L 5 0.001 for L>H 5 8 The 1955 Soviet Code of Practice gives the following allowable values: Type of building L ,H Multistory buildings and civil dwellings <3 0.0003 (for sand) 0.0004 (for clay) >5 0.0005 (for sand) 0.0007 (for clay) One-story mills D ,L 0.001 (for sand and clay) Bjerrum (1963) recommended the following limiting angular distortion, bmax for various structures: Category of potential damage Safe limit for flexible brick wall (L>H . 4) Danger of structural damage to most buildings Cracking of panel and brick walls Visible tilting of high rigid buildings First cracking of panel walls Safe limit for no cracking of building Danger to frames with diagonals bmax 1>150 1>150 1>150 1>250 1>300 1>500 1>600 If the maximum allowable values of bmax are known, the magnitude of the allowable ST(max) can be calculated with the use of the foregoing correlations. The European Committee for Standardization has also provided limiting values for serviceability and the maximum accepted foundation movements. (See Table 5.15.) Problems 285 Table 5.15 Recommendations of European Committee for Standardization on Differential Settlement Parameters Item Limiting values for serviceability (European Committee for Standardization, 1994a) Maximum acceptable foundation movement (European Committee for Standardization, 1994b) Parameter ST DST b ST DST b Magnitude Comments 25 mm 50 mm 5 mm 10 mm 20 mm 1>500 50 20 <1>500 Isolated shallow foundation Raft foundation Frames with rigid cladding Frames with flexible cladding Open frames — Isolated shallow foundation Isolated shallow foundation — Problems A flexible circular area is subjected to a uniformly distributed load of 150 kN> m2. The diameter of the loaded area is 2 m. Determine the stress increase in a soil mass at a point located 3 m below the center of the loaded area. a. by using Eq. (5.3) b. by using Eq. (5.28). Use s 5 0 5.2 Refer to Figure 5.4, which shows a flexible rectangular area. Given: B1 5 1.22 m, B2 5 1.83 m, L1 5 2.44 m, and L2 5 3.05 m. If the area is subjected to a uniform load of 3000 lb> ft2, determine the stress increase at a depth of 3.05 m located immediately below point O. 5.3 Repeat Problem 5.2 with the following: B1 5 1.22 m, B2 5 3.05 m, L1 5 2.44 m, L2 5 3.66 m, and the uniform load on the flexible area 5 120 kN/m2. Determine the stress increase below point O at a depth of 6.1 m. Use Eq. (5.29) and s 5 0. 5.4 Using Eq. (5.10), determine the stress increase (Ds) at z 5 3.05 m below the center of the area described in Problem 5.2. 5.5 Refer to Figure P5.5. Using the procedure outlined in Section 5.5, determine the average stress increase in the clay layer below the center of the foundation due to the net foundation load of 900 kN. 5.6 Solve Problem 5.5 using the 2:1 method [Eqs. (5.14) and (5.88)]. 5.7 Figure P5.7 shows an embankment load on a silty clay soil layer. Determine the stress increase at points A, B, and C, which are located at a depth of 5 m below the ground surface. 5.8 A planned flexible load area (see Figure P5.8) is to be 2 m 3 3.2 m and carries a uniformly distributed load of 210 kN> m2. Estimate the elastic settlement below the center of the loaded area. Assume that Df 5 1.6 m and H 5 `. Use Eq. (5.33). 5.9 Redo Problem 5.8 assuming that Df 5 1.2 m and H 5 4 m. 5.10 Consider a flexible foundation measuring 1.52 m 3.05 m in a plan on a soft saturated clay (s 0.5). The depth of the foundation is 1.22 m below the ground 5.1 286 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement 900 kN (net load) Sand 15.7 kN/m3 1.52 m 1.83 m 1.83 m 1.22 m Groundwater table Sand sat 19.24 kN/m3 3.05 m sat 19.24 kN/m3 eo 0.8 Cc 0.25 Cs 0.06 Preconsolidation pressure 100 kN/m2 Figure P5.5 6m Center line 1V:2H 1V:2H 10 m γ = 17 kN/m3 5m C B Df A Figure P5.7 210 kN/m2 2 m 3.2 m Silty sand Es 8500 kN/m2 s 0.3 H Rock Figure P5.8 surface. A rigid rock layer is located at 12.2 m below the bottom of the foundation. Given: qo 144 kN/m2 and, for the clay, Es 12,938 kN/m2. Determine the average elastic settlement of the foundation. Use Eq. (5.30). 5.11 Figure 5.16 shows a foundation of 3.05 m 3 1.91 m resting on a sand deposit. The net load per unit area at the level of the foundation, qo, is 144 kN> m2. For the sand, ms 5 0.3, Es 5 22,080 kN> m2, Df 5 0.76 m and H 5 9.76 m. Assume that the foundation is rigid, and determine the elastic settlement the foundation would undergo. Use Eqs. (5.33) and (5.41). Problems 287 5.12 Repeat Problems 5.11 for a foundation of size 5 1.8 m 3 1.8 m and with qo 5 190 kN> m2, Df 5 1.0 m, H 5 15 m; and soil conditions of ms 5 0.4, Es 5 15,400 kN> m2, and g 5 17 kN> m3. 5.13 For a shallow foundation supported by a silty clay, as shown in Figure 5.17, the following are given: Length, L 5 2 m Width, B 5 1 m Depth of foundation, Df 5 1 m Thickness of foundation, t 5 0.23 m Load per unit area, qo 5 190 kN> m2 Ef 5 15 3 106 kN> m2 The silty clay soil has the following properties: H52m ms 5 0.4 Eo 5 9000 kN> m2 k 5 500 kN> m2> m Using Eq. (5.46), estimate the elastic settlement of the foundation. 5.14 A plan calls for a square foundation measuring 3 m 3 3 m, supported by a layer of sand. (See Figure 5.17.) Led Df 5 1.5 m, t 5 0.25 m, Eo 5 16,000 kN> m2, k 5 400 kN> m2> m, ms 5 0.3, H 5 20 m, Ef 5 15 3 106 kN> m2, and qo 5 150 kN> m2. Calculate the elastic settlement. Use Eq. (5.46). 5.15 Solve Problem 5.11 with Eq. (5.49). For the correction factor C2-, use a time of 5 years for creep and, for the unit weight of soil, use g 5 18.08 kN> m3. 5.16 A continuous foundation on a deposit of sand layer is shown in Figure P5.16 along with the variation of the modulus of elasticity of the soil (Es). Assuming g 5 18 kN> m3 and C2 for 10 years, calculate the elastic settlement of the foundation using the strain influence factor method. 1.5 m q 195 kN/m2 0 Es 6000 2.5 m Sand Es(kN/m2) 2 Es 12,000 8 Es 10,000 14 Depth (m) Figure P5.16 288 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement 5.17 Following are the results of standard penetration tests in a granular soil deposit. Depth (m) Standard penetration number, N60 1.5 3.0 4.5 6.0 7.5 11 10 12 9 14 What will be the net allowable bearing capacity of a foundation planned to be 1.83 m 3 1.83 m? Let Df 5 6.91 m and the allowable settlement 5 25 mm, and use the relationships presented in Eq. (5.60). 5.18 A shallow foundation measuring 1.2 m 3 1.2 m in plan is to be constructed over a normally consolidated sand layer. Given: Df 5 1 m, N60 increases with depth, N60 (in the depth of stress influence) 5 8, and qnet 5 210 kN> m2. Estimate the elastic settlement using Burland and Burbidge’s method. 5.19 Refer to Figure 5.25. For a foundation on a layer of sand, given: B 1.52 m, L 3.05 m, d 1.52 m, 26.6°, e 0.152 m, and 109. The Pressuremeter testing at the site produced a mean Pressuremeter curve for which the pp(m) versus R/Ro points are as follow: R/Ro (1) pp(m) (lb/in2) (2) 0.002 0.004 0.008 0.012 0.024 0.05 0.08 0.1 0.2 49.68 166.98 224.94 292.56 475.41 870.09 1225.97 1452.45 2550.24 What should be the magnitude of Qo for a settlement (center) of 25 mm? 5.20 Estimate the consolidation settlement of the clay layer shown in Figure P5.5 using the results of Problem 5.5. 5.21 Estimate the consolidation settlement of the clay layer shown in Figure P5.5 using the results of Problem 5.6. References AHLVIN, R. G., and ULERY, H. H. (1962). Tabulated Values of Determining the Composite Pattern of Stresses, Strains, and Deflections beneath a Uniform Load on a Homogeneous Half Space. Highway Research Board Bulletin 342, pp. 1–13. BJERRUM, L. (1963). “Allowable Settlement of Structures,” Proceedings, European Conference on Soil Mechanics and Foundation Engineering, Wiesbaden, Germany, Vol. III, pp. 135 –137. References 289 BOUSSINESQ, J. (1883). Application des Potentials á L’Étude de L’Équilibre et du Mouvement des Solides Élastiques, Gauthier-Villars, Paris. BOWLES, J. E. (1987). “Elastic Foundation Settlement on Sand Deposits,” Journal of Geotechnical Engineering, ASCE, Vol. 113, No. 8, pp. 846– 860. BOWLES, J. E. (1977). Foundation Analysis and Design, 2d ed., McGraw-Hill, New York. BRIAUD, J. L. (2007). “Spread Footings in Sand: Load Settlement Curve Approach,” Journal of Geotechnical and Geoenvironmental Engineering, American Society of Civil Engineers, Vol. 133, No. 8, pp. 905–920. BURLAND, J. B., and BURBIDGE, M. C. (1985). “Settlement of Foundations on Sand and Gravel,” Proceedings, Institute of Civil Engineers, Part I, Vol. 7, pp. 1325–1381. CHRISTIAN, J. T., and CARRIER, W. D. (1978). “Janbu, Bjerrum, and Kjaernsli’s Chart Reinterpreted,” Canadian Geotechnical Journal, Vol. 15, pp. 124–128. DAS, B. (2008). Advanced Soil Mechanics, 3d ed., Taylor and Francis, London. DUNCAN, J. M., and BUCHIGNANI, A. N. (1976). An Engineering Manual for Settlement Studies, Department of Civil Engineering, University of California, Berkeley. EUROPEAN COMMITTEE FOR STANDARDIZATION (1994a). Basis of Design and Actions on Structures, Eurocode 1, Brussels, Belgium. EUROPEAN COMMITTEE FOR STANDARDIZATION (1994b). Geotechnical Design, General Rules—Part 1, Eurocode 7, Brussels, Belgium. FOX, E. N. (1948). “The Mean Elastic Settlement of a Uniformly Loaded Area at a Depth below the Ground Surface,” Proceedings, 2nd International Conference on Soil Mechanics and Foundation Engineering, Rotterdam, Vol. 1, pp. 129–132. GRIFFITHS, D. V. (1984). “A Chart for Estimating the Average Vertical Stress Increase in an Elastic Foundation below a Uniformly Loaded Rectangular Area,” Canadian Geotechnical Journal, Vol. 21, No. 4, pp. 710–713. JANBU, N., BJERRUM, L., and KJAERNSLI, B. (1956). “Veiledning vedlosning av fundamentering— soppgaver,” Publication No. 18, Norwegian Geotechnical Institute, pp. 30–32. LEONARDS, G. A. (1976). Estimating Consolidation Settlement of Shallow Foundations on Overconsolidated Clay, Special Report No. 163, Transportation Research Board, Washington, D.C., pp. 13–16. MAYNE, P. W., and POULOS, H. G. (1999). “Approximate Displacement Influence Factors for Elastic Shallow Foundations,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 125, No. 6, pp. 453– 460. MESRI, G. (1973). “Coefficient of Secondary Compression,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 99, No. SM1, pp. 122–137. MESRI, G., and GODLEWSKI, P. M. (1977). “Time and Stress—Compressibility Interrelationship,” Journal of Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 103, No. GT5, pp. 417–430. MEYERHOF, G. G. (1956). “Penetration Tests and Bearing Capacity of Cohesionless Soils,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 82, No. SM1, pp. 1–19. MEYERHOF, G. G. (1965). “Shallow Foundations,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 91, No. SM2, pp. 21–31. NEWMARK, N. M. (1935). Simplified Computation of Vertical Pressure in Elastic Foundation, Circular 24, University of Illinois Engineering Experiment Station, Urbana, IL. OSTERBERG, J. O. (1957). “Influence Values for Vertical Stresses in Semi-Infinite Mass Due to Embankment Loading,” Proceedings, Fourth International Conference on Soil Mechanics and Foundation Engineering, London, Vol. 1, pp. 393 –396. POLSHIN, D. E., and TOKAR, R. A. (1957). “Maximum Allowable Nonuniform Settlement of Structures,” Proceedings, Fourth International Conference on Soil Mechanics and Foundation Engineering, London, Vol. 1, pp. 402–405. SALGADO, R. (2008). The Engineering of Foundations, McGraw-Hill, New York. 290 Chapter 5: Shallow Foundations: Allowable Bearing Capacity and Settlement SCHMERTMANN, J. H., HARTMAN, J. P., and BROWN, P. R. (1978). “Improved Strain Influence Factor Diagrams,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 104, No. GT8, pp. 1131–1135. SKEMPTON, A. W., and BJERRUM, L. (1957). “ A Contribution to Settlement Analysis of Foundations in Clay,” Geotechnique, London, Vol 7, p. 178. SKEMPTON, A. W., and MCDONALD, D. M. (1956). “The Allowable Settlement of Buildings,” Proceedings of Institute of Civil Engineers, Vol. 5, Part III, p. 727. STEINBRENNER, W. (1934). “Tafeln zur Setzungsberechnung,” Die Strasse, Vol. 1, pp. 121–124. TERZAGHI, K., PECK, R. B., and MESRI, G. (1996). Soil Mechanics in Engineering Practice, 3rd Edition, Wiley, New York. WESTERGAARD, H. M. (1938). “A Problem of Elasticity Suggested by a Problem in Soil Mechanics: Soft Material Reinforced by Numerous Strong Horizontal Sheets,” Contributions to the Mechanics of Solids, Dedicated to Stephen Timoshenko, pp. 268–277, MacMillan, New York. 6 6.1 Mat Foundations Introduction Under normal conditions, square and rectangular footings such as those described in Chapters 3 and 4 are economical for supporting columns and walls. However, under certain circumstances, it may be desirable to construct a footing that supports a line of two or more columns. These footings are referred to as combined footings. When more than one line of columns is supported by a concrete slab, it is called a mat foundation. Combined footings can be classified generally under the following categories: a. Rectangular combined footing b. Trapezoidal combined footing c. Strap footing Mat foundations are generally used with soil that has a low bearing capacity. A brief overview of the principles of combined footings is given in Section 6.2, followed by a more detailed discussion on mat foundations. 6.2 Combined Footings Rectangular Combined Footing In several instances, the load to be carried by a column and the soil bearing capacity are such that the standard spread footing design will require extension of the column foundation beyond the property line. In such a case, two or more columns can be supported on a single rectangular foundation, as shown in Figure 6.1. If the net allowable soil pressure is known, the size of the foundation (B 3 L) can be determined in the following manner: a. Determine the area of the foundation A5 Q1 1 Q2 qnet(all) (6.1) where Q1 , Q2 5 column loads qnet(all) 5 net allowable soil bearing capacity 291 292 Chapter 6: Mat Foundations Q1 Q2 L2 L1 X L3 Q1 Q2 Section B qnet(all) /unit length L Property line B Plan Figure 6.1 Rectangular combined footing b. Determine the location of the resultant of the column loads. From Figure 6.1, X5 Q2L3 Q1 1 Q2 (6.2) c. For a uniform distribution of soil pressure under the foundation, the resultant of the column loads should pass through the centroid of the foundation. Thus, L 5 2(L2 1 X) (6.3) where L 5 length of the foundation. d. Once the length L is determined, the value of L1 can be obtained as follows: L1 5 L 2 L2 2 L3 (6.4) Note that the magnitude of L2 will be known and depends on the location of the property line. e. The width of the foundation is then B5 A L (6.5) Trapezoidal Combined Footing Trapezoidal combined footing (see Figure 6.2) is sometimes used as an isolated spread foundation of columns carrying large loads where space is tight. The size of the foundation that will uniformly distribute pressure on the soil can be obtained in the following manner: a. If the net allowable soil pressure is known, determine the area of the foundation: Q1 1 Q2 A5 qnet(all) 6.2 Combined Footings Q1 Q2 L2 L1 X L3 Q1 293 Q2 B2 qnet(all) /unit length B1 qnet(all) /unit length Section Property line B1 B2 L Plan Figure 6.2 Trapezoidal combined footing From Figure 6.2, A5 B1 1 B2 L 2 (6.6) b. Determine the location of the resultant for the column loads: X5 Q2L3 Q1 1 Q2 c. From the property of a trapezoid, X 1 L2 5 ¢ B1 1 2B2 L ≤ B1 1 B2 3 (6.7) With known values of A, L, X, and L2 , solve Eqs. (6.6) and (6.7) to obtain B1 and B2 . Note that, for a trapezoid, L L , X 1 L2 , 3 2 294 Chapter 6: Mat Foundations Section Section Strap Strap Strap Strap Plan (a) Plan (b) Wall Section Strap Strap Plan (c) Figure 6.3 Cantilever footing—use of strap beam Cantilever Footing Cantilever footing construction uses a strap beam to connect an eccentrically loaded column foundation to the foundation of an interior column. (See Figure 6.3). Cantilever footings may be used in place of trapezoidal or rectangular combined footings when the allowable soil bearing capacity is high and the distances between the columns are large. 6.3 Common Types of Mat Foundations The mat foundation, which is sometimes referred to as a raft foundation, is a combined footing that may cover the entire area under a structure supporting several columns and walls. Mat foundations are sometimes preferred for soils that have low load-bearing capacities, but that will have to support high column or wall loads. Under some conditions, spread footings would have to cover more than half the building area, and mat foundations might be more economical. Several types of mat foundations are used currently. Some of the common ones are shown schematically in Figure 6.4 and include the following: 1. Flat plate (Figure 6.4a). The mat is of uniform thickness. 2. Flat plate thickened under columns (Figure 6.4b). 3. Beams and slab (Figure 6.4c). The beams run both ways, and the columns are located at the intersection of the beams. 4. Flat plates with pedestals (Figure 6.4d). 5. Slab with basement walls as a part of the mat (Figure 6.4e). The walls act as stiffeners for the mat. 6.3 Common Types of Mat Foundations 295 Section Section Section Plan Plan Plan (a) (b) (c) Section Section Plan Plan (e) (d) Figure 6.4 Common types of mat foundations Mats may be supported by piles, which help reduce the settlement of a structure built over highly compressible soil. Where the water table is high, mats are often placed over piles to control buoyancy. Figure 6.5 shows the difference between the depth Df and the width B of isolated foundations and mat foundations. Figure 6.6 shows a flat-plate mat foundation under construction. Df B Df B Figure 6.5 Comparison of isolated foundation and mat foundation (B 5 width, Df 5 depth) 296 Chapter 6: Mat Foundations Figure 6.6 A flat plate mat foundation under construction (Courtesy of Dharma Shakya, Geotechnical Solutions, Inc., Irvine, California) 6.4 Bearing Capacity of Mat Foundations The gross ultimate bearing capacity of a mat foundation can be determined by the same equation used for shallow foundations (see Section 3.6), or qu 5 crNcFcsFcdFci 1 qNqFqsFqdFqi 1 12gBNgFgsFgdFgi [Eq. (3.19)] (Chapter 3 gives the proper values of the bearing capacity factors, as well as the shape depth, and load inclination factors.) The term B in Eq. (3.19) is the smallest dimension of the mat. The net ultimate capacity of a mat foundation is qnet(u) 5 qu 2 q [Eq. (3.14)] A suitable factor of safety should be used to calculate the net allowable bearing capacity. For mats on clay, the factor of safety should not be less than 3 under dead load or maximum live load. However, under the most extreme conditions, the factor of safety should be at least 1.75 to 2. For mats constructed over sand, a factor of safety of 3 should normally be used. Under most working conditions, the factor of safety against bearing capacity failure of mats on sand is very large. For saturated clays with f 5 0 and a vertical loading condition, Eq. (3.19) gives qu 5 cuNcFcsFcd 1 q (6.8) 6.4 Bearing Capacity of Mat Foundations 297 where cu 5 undrained cohesion. (Note: Nc 5 5.14, Nq 5 1, and Ng 5 0.) From Table 3.4, for f 5 0, Fcs 5 1 1 B Nq B 1 0.195B ¢ ≤ 5 1 1 ¢ ≤¢ ≤ 511 L Nc L 5.14 L and Fcd 5 1 1 0.4 ¢ Df B ≤ Substitution of the preceding shape and depth factors into Eq. (6.8) yields qu 5 5.14cu ¢ 1 1 Df 0.195B ≤ ¢1 1 0.4 ≤ 1 q L B (6.9) Hence, the net ultimate bearing capacity is qnet(u) 5 qu 2 q 5 5.14cu ¢ 1 1 Df 0.195B ≤ ¢ 1 1 0.4 ≤ L B (6.10) For FS 5 3, the net allowable soil bearing capacity becomes qnet(all) 5 qu(net) FS 5 1.713cu ¢ 1 1 Df 0.195B ≤ ¢ 1 1 0.4 ≤ L B (6.11) The net allowable bearing capacity for mats constructed over granular soil deposits can be adequately determined from the standard penetration resistance numbers. From Eq. (5.64), for shallow foundations, qnet (kN>m2 ) 5 N60 B 1 0.3 2 Se ¢ ≤ Fd ¢ ≤ 0.08 B 25 [Eq. (5.64)] where N60 5 standard penetration resistance B 5 width (m) Fd 5 1 1 0.33(Df>B) < 1.33 Se 5 settlement, (mm) When the width B is large, the preceding equation can be approximated as N60 Se F ¢ ≤ 0.08 d 25 Df N60 Se (mm) 5 B1 1 0.33¢ ≤ R B R 0.08 B 25 qnet (kN>m2 ) 5 < 16.63N60 B Se (mm) R 25 (6.12) 298 Chapter 6: Mat Foundations In English units, Eq. (6.12) may be expressed as qnet(all) (kip>ft 2 ) 5 0.25N60 B1 1 0.33¢ < 0.33N60 3Se (in.) 4 Df B ≤ R 3Se (in.)4 (6.13) Generally, shallow foundations are designed for a maximum settlement of 25 mm and a differential settlement of about 19 mm. However, the width of the raft foundations are larger than those of the isolated spread footings. As shown in Table 5.3, the depth of significant stress increase in the soil below a foundation depends on the width of the foundation. Hence, for a raft foundation, the depth of the zone of influence is likely to be much larger than that of a spread footing. Thus, the loose soil pockets under a raft may be more evenly distributed, resulting in a smaller differential settlement. Accordingly, the customary assumption is that, for a maximum raft settlement of 50 mm, the differential settlement would be 19 mm. Using this logic and conservatively assuming that Fd 5 1, we can respectively approximate Eqs. (6.12) and (6.13) as qnet(all) 5 qnet (kN>m2 ) < 25N60 (6.14a) The net allowable pressure applied on a foundation (see Figure 6.7) may be expressed as q5 Q 2 gDf A where Q 5 dead weight of the structure and the live load A 5 area of the raft In all cases, q should be less than or equal to allowable qnet. Unit weight Df Q Figure 6.7 Definition of net pressure on soil caused by a mat foundation (6.15) 6.5 Differential Settlement of Mats 299 Example 6.1 Determine the net ultimate bearing capacity of a mat foundation measuring 15 m 3 10 m on a saturated clay with cu 5 95 kN>m2, f 5 0, and Df 5 2 m. Solution From Eq. (6.10), qnet(u) 5 5.14cu B1 1 ¢ Df 0.195B ≤ R B1 1 0.4 R L B 5 (5.14) (95) B1 1 ¢ 0.195 3 10 0.4 3 2 ≤ R B1 1 ¢ ≤R 15 10 5 595.9 kN , m2 ■ Example 6.2 What will be the net allowable bearing capacity of a mat foundation with dimensions of 15 m 3 10 m constructed over a sand deposit? Here, Df 5 2 m, the allowable settlement is 25 mm, and the average penetration number N60 5 10. Solution From Eq. (6.12), qnet(all) 5 Df N60 Se Se B1 1 0.33¢ ≤ R ¢ ≤ < 16.63N60 ¢ ≤ 0.08 B 25 25 qnet(all) 5 10 0.33 3 2 25 B1 1 R ¢ ≤ 5 133.25 kN , m2 0.08 10 25 or 6.5 ■ Differential Settlement of Mats In 1988, the American Concrete Institute Committee 336 suggested a method for calculating the differential settlement of mat foundations. According to this method, the rigidity factor Kr is calculated as ErIb (6.16) Kr 5 EsB3 where Er 5 Es 5 B5 Ib 5 modulus of elasticity of the material used in the structure modulus of elasticity of the soil width of foundation moment of inertia of the structure per unit length at right angles to B The term ErIb can be expressed as ah3 ErIb 5 Er ¢IF 1 a Irb 1 a ≤ 12 (6.17) 300 Chapter 6: Mat Foundations where ErIb 5 flexural rigidity of the superstructure and foundation per unit length at right angles to B SErI br 5 flexural rigidity of the framed members at right angles to B S(Erah3>12) 5 flexural rigidity of the shear walls a 5 shear wall thickness h 5 shear wall height ErIF 5 flexibility of the foundation Based on the value of Kr , the ratio (d) of the differential settlement to the total settlement can be estimated in the following manner: 1. If Kr . 0.5, it can be treated as a rigid mat, and d 5 0. 2. If Kr 5 0.5, then d < 0.1. 3. If Kr 5 0, then d 5 0.35 for square mats (B>L 5 1) and d 5 0.5 for long foundations (B>L 5 0). 6.6 Field Settlement Observations for Mat Foundations Several field settlement observations for mat foundations are currently available in the literature. In this section, we compare the observed settlements for some mat foundations constructed over granular soil deposits with those obtained from Eqs. (6.12) and (6.13). Meyerhof (1965) compiled the observed maximum settlements for mat foundations constructed on sand and gravel, as listed in Table 6.1. In Eq. (6.12), if the depth factor, 1 1 0.33(Df>B), is assumed to be approximately unity, then Se (mm) < 2qnet(all) N60 (6.18) From the values of qnet(all) and N60 given in Columns 6 and 5, respectively, of Table 6.1, the magnitudes of Se were calculated and are given in Column 8. Column 9 of Table 6.1 gives the ratios of calculated to measured values of Se . These ratios vary from about 0.79 to 3.39. Thus, calculating the net allowable bearing capacity with the use of Eq. (6.12) or (6.13) will yield safe and conservative values. 6.7 Compensated Foundation Figure 6.7 and Eq. (6.15) indicate that the net pressure increase in the soil under a mat foundation can be reduced by increasing the depth Df of the mat. This approach is generally referred to as the compensated foundation design and is extremely useful when structures are to be built on very soft clays. In this design, a deeper basement is made below the higher portion of the superstructure, so that the net pressure increase in soil at any depth is relatively uniform. (See Figure 6.8.) From Eq. (6.15) and Figure 6.7, the net average applied pressure on soil is q5 Q 2 gDf A 301 T. Edison São Paulo, Brazil Banco do Brazil São Paulo, Brazil Iparanga São Paulo, Brazil C.B.I., Esplanda São Paulo, Brazil Riscala São Paulo, Brazil Thyssen Düsseldorf, Germany Ministry Düsseldorf, Germany Chimney Cologne, Germany 2 3 4 5 6 7 8 Structure (2) 1 Case No. (1) Schultze (1962) Schultze (1962) Schultze (1962) Vargas (1948) Vargas (1961) Vargas (1948) Rios and Silva (1948); Vargas (1961) Rios and Silva (1948) Reference (3) 20.42 15.85 22.55 3.96 14.63 9.14 22.86 18.29 B m (4) 10 20 25 20 22 9 18 15 Average N60 (5) 172.4 220.2 239.4 229.8 383.0 304.4 239.4 229.8 qnet(all) kN , m2 (6) 10.16 20.32 24.13 12.7 27.94 35.56 27.94 15.24 Observed maximum settlement, Se mm (7) 34.48 22.02 19.15 22.98 34.82 67.64 26.6 30.64 Calculated maximum settlement, Se mm (8) 3.39 1.08 0.79 1.81 1.25 1.9 0.95 2.01 calculated Se observed Se (9) Table 6.1 Settlement of Mat Foundations on Sand and Gravel (Based on Meyerhof, 1965) (Based on Meyerhof, G. G., (1965). “Shallow Foundations,” Journal of the Soil Mechanics and Foundation Engineering Divsion, American Society of Civil Engineers, Vol. 91, No. 2, pp. 21–31, Table 1. With permission from ASCE.) 302 Chapter 6: Mat Foundations Figure 6.8 Compensated foundation For no increase in the net pressure on soil below a mat foundation, q should be zero. Thus, Df 5 Q Ag (6.19) This relation for Df is usually referred to as the depth of a fully compensated foundation. The factor of safety against bearing capacity failure for partially compensated foundations (i.e., Df , Q>Ag) may be given as FS 5 qnet(u) q 5 qnet(u) Q 2 gDf A (6.20) where qnet(u) 5 net ultimate bearing capacity. For saturated clays, the factor of safety against bearing capacity failure can thus be obtained by substituting Eq. (6.10) into Eq. (6.20): 5.14cu ¢ 1 1 FS 5 Df 0.195B ≤ ¢1 1 0.4 ≤ L B Q 2 gDf A (6.21) Example 6.3 The mat shown in Figure 6.7 has dimensions of 18.3 m 3 30.5 m. The total dead and live load on the mat is 111 3 103 kN kip. The mat is placed over a saturated clay having a unit weight of 18.87 kN>m3 and cu 5 134 kN>m2. Given that Df 5 1.52 m, determine the factor of safety against bearing capacity failure. 6.7 Compensated Foundation 303 Solution From Eq. (6.21), the factor of safety 5.14cu ¢1 1 FS 5 Df 0.195B ≤ ¢ 1 1 0.4 ≤ L B Q 2 gDf A We are given that cu 5 134 kN>m2, Df 5 1.52 m, B 5 18.3 m, L 5 30.5 m, and g 5 18.87 kN>m3. Hence, (5.14) (134) B1 1 FS 5 (0.195) (18.3) 1.52 R B1 1 0.4¢ ≤R 30.5 18.3 111 3 103 kN ¢ ≤ 2 (18.87) (1.52) 18.3 3 30.5 5 4.66 ■ Example 6.4 Consider a mat foundation 30 m 3 40 m in plan, as shown in Figure 6.9. The total dead load and live load on the raft is 200 3 103 kN. Estimate the consolidation settlement at the center of the foundation. Solution From Eq. (1.61) Sc(p) 5 CcHc sor 1 Dsav r log ¢ ≤ 1 1 eo sor 6 sor 5 (3.67)(15.72) 1 (13.33) (19.1 2 9.81) 1 (18.55 2 9.81) < 208 kN>m2 2 Hc 5 6 m Cc 5 0.28 eo 5 0.9 For Q 5 200 3 103 kN, the net load per unit area is q5 Q 200 3 103 2 gDf 5 2 (15.72) (2) < 135.2 kN>m2 A 30 3 40 r we refer to Section 5.5. The loaded area can be divided into In order to calculate Dsav four areas, each measuring 15 m 3 20 m. Now using Eq. (5.19), we can calculate the average stress increase in the clay layer below the corner of each rectangular area, or Dsav(H r 2>H1) 5 qo B H2Ia(H2) 2 H1Ia(H1) 5 135.2B R H2 2 H1 (1.67 1 13.3 1 6)Ia(H2) 2 (1.67 1 13.33)Ia(H1) 6 R 304 Chapter 6: Mat Foundations Q 30 m 40 m 2m 1.67 m Sand ␥ 15.72 kN/m3 Groundwater table z Sand ␥sat 19.1 kN/m3 13.33 m Normally consolidated clay ␥sat 18.55 kN/m3 Cc 0.28; eo 0.9 6m Sand Figure 6.9 Consolidation settlement under a mat foundation For Ia(H2) , B 15 5 5 0.71 H2 1.67 1 13.33 1 6 L 20 n2 5 5 5 0.95 H2 21 m2 5 From Fig. 5.7, for m2 5 0.71 and n2 5 0.95, the value of Ia(H2) is 0.21. Again, for Ia(H1), B 15 m2 5 5 51 H1 15 L 20 n2 5 5 5 1.33 H1 15 From Figure 5.7, Ia(H1) 5 0.225, so Dsav(H r 2>H1) 5 135.2B (21) (0.21) 2 (15) (0.225) R 5 23.32 kN>m2 6 So, the stress increase below the center of the 30 m 3 40 m area is (4) (23.32) 5 93.28 kN>m2. Thus Sc(p) 5 6.8 (0.28) (6) 208 1 93.28 log ¢ ≤ 5 0.142 m 1 1 0.9 208 5 142 mm ■ Structural Design of Mat Foundations The structural design of mat foundations can be carried out by two conventional methods: the conventional rigid method and the approximate flexible method. Finite-difference and finite-element methods can also be used, but this section covers only the basic concepts of the first two design methods. 6.8 Structural Design of Mat Foundations 305 Conventional Rigid Method The conventional rigid method of mat foundation design can be explained step by step with reference to Figure 6.10: Step 1. Figure 6.10a shows mat dimensions of L 3 B and column loads of Q1 , Q2 , Q3 , c. Calculate the total column load as Q 5 Q1 1 Q2 1 Q3 1 c (6.22) Step 2. Determine the pressure on the soil, q, below the mat at points A, B, C, D, c, by using the equation q5 Myx Q Mxy 6 6 A Iy Ix (6.23) where A 5 BL Ix 5 (1>12)BL3 5 moment of inertia about the x-axis Iy 5 (1>12)LB3 5 moment of inertia about the y-axis Mx 5 moment of the column loads about the x-axis 5 Qey My 5 moment of the column loads about the y-axis 5 Qex The load eccentricities, ex and ey , in the x and y directions can be determined by using (xr, yr) coordinates: xr 5 Q1x1r 1 Q2x2r 1 Q3x3r 1 c Q (6.24) and ex 5 xr 2 B 2 (6.25) Similarly, yr 5 Q1y1r 1 Q2y2r 1 Q3y3r 1 c Q (6.26) and ey 5 yr 2 L 2 (6.27) Step 3. Compare the values of the soil pressures determined in Step 2 with the net allowable soil pressure to determine whether q < qall(net) . Step 4. Divide the mat into several strips in the x and y directions. (See Figure 6.10). Let the width of any strip be B1 . y y B1 A B1 B1 B Q9 B1 D C Q11 Q10 Q12 B1 ex B1 ey L E J Q5 Q6 Q7 Q8 Q1 Q2 Q3 Q4 H G x B1 I x F B (a) FQ1 FQ2 I FQ4 FQ3 H G F B1 qav(modified) unit length B (b) Edge of mat L L d/2 d/2 d/2 L Edge of mat d/2 b o 2L L Edge of mat d/2 d/2 d/2 L b o L L (c) Figure 6.10 Conventional rigid mat foundation design 306 d/2 L d/2 b o 2(L L) L 307 6.8 Structural Design of Mat Foundations Step 5. Draw the shear, V, and the moment, M, diagrams for each individual strip (in the x and y directions). For example, the average soil pressure of the bottom strip in the x direction of Figure 6.10a is qav < qI 1 qF 2 (6.28) where qI and qF 5 soil pressures at points I and F, as determined from Step 2. The total soil reaction is equal to qavB1B. Now obtain the total column load on the strip as Q1 1 Q2 1 Q3 1 Q4 . The sum of the column loads on the strip will not equal qavB1B, because the shear between the adjacent strips has not been taken into account. For this reason, the soil reaction and the column loads need to be adjusted, or Average load 5 qavB1B 1 (Q1 1 Q2 1 Q3 1 Q4 ) 2 (6.29) Now, the modified average soil reaction becomes qav(modified) 5 qav ¢ average load ≤ qavB1B (6.30) and the column load modification factor is F5 average load Q1 1 Q2 1 Q3 1 Q4 (6.31) So the modified column loads are FQ1 , FQ2 , FQ3 , and FQ4 . This modified loading on the strip under consideration is shown in Figure 6.10b. The shear and the moment diagram for this strip can now be drawn, and the procedure is repeated in the x and y directions for all strips. Step 6. Determine the effective depth d of the mat by checking for diagonal tension shear near various columns. According to ACI Code 318-95 (Section 11.12.2.1c, American Concrete Institute, 1995), for the critical section, U 5 bod3f(0.34)"fcr 4 (6.32) where U 5 factored column load (MN), or (column load) 3 (load factor) f 5 reduction factor 5 0.85 fcr 5 compressive strength of concrete at 28 days (MN>m2 ) The units of bo and d in Eq. (6.32a) are in meters. The expression for bo in terms of d, which depends on the location of the column with respect to the plan of the mat, can be obtained from Figure 6.10c. 308 Chapter 6: Mat Foundations Step 7. From the moment diagrams of all strips in one direction (x or y), obtain the maximum positive and negative moments per unit width (i.e., Mr 5 M>B1). Step 8. Determine the areas of steel per unit width for positive and negative reinforcement in the x and y directions. We have Mu 5 (Mr) (load factor) 5 fA sfy ¢d 2 a ≤ 2 (6.33) and a5 A sfy 0.85fcr b (6.34) where As 5 fy 5 Mu 5 f5 area of steel per unit width yield stress of reinforcement in tension factored moment 0.9 5 reduction factor Examples 6.5 and 6.6 illustrate the use of the conventional rigid method of mat foundation design. Approximate Flexible Method In the conventional rigid method of design, the mat is assumed to be infinitely rigid. Also, the soil pressure is distributed in a straight line, and the centroid of the soil pressure is coincident with the line of action of the resultant column loads. (See Figure 6.11a.) In the approximate flexible method of design, the soil is assumed to be equivalent to an infinite number of elastic springs, as shown in Figure 6.11b. This assumption is sometimes referred to as the Winkler foundation. The elastic constant of these assumed springs is referred to as the coefficient of subgrade reaction, k. To understand the fundamental concepts behind flexible foundation design, consider a beam of width B1 having infinite length, as shown in Figure 6.11c. The beam is subjected to a single concentrated load Q. From the fundamentals of mechanics of materials, M 5 EFIF d2z dx2 (6.35) where M 5 moment at any section EF 5 modulus of elasticity of foundation material IF 5 moment of inertia of the cross section of the beam 5 ( 121 )B1h3 (see Figure 6.11c). However, dM 5 shear force 5 V dx and dV 5 q 5 soil reaction dx 6.8 Structural Design of Mat Foundations 309 Q Q1 Q3 Q2 Resultant of soil pressure (a) Q2 Q1 (b) Point load A B1 x h Section at A A q A z (c) Figure 6.11 (a) Principles of design by conventional rigid method; (b) principles of approximate flexible method; (c) derivation of Eq. (6.39) for beams on elastic foundation Hence, d2M 5q dx2 (6.36) Combining Eqs. (6.35) and (6.36) yields EFIF d4z 5q dx4 However, the soil reaction is q 5 2zkr (6.37) 310 Chapter 6: Mat Foundations where z 5 deflection kr 5 kB1 k 5 coefficient of subgrade reaction (kN>m3 or lb>in3 ) So, EFIF d4z 5 2zkB1 dx4 (6.38) Solving Eq. (6.38) yields z 5 e2ax (Ar cos bx 1 As sin bx) (6.39) where Ar and As are constants and b5 B1k Å 4EFIF 4 (6.40) The unit of the term b, as defined by the preceding equation, is (length) 21. This parameter is very important in determining whether a mat foundation should be designed by the conventional rigid method or the approximate flexible method. According to the American Concrete Institute Committee 336 (1988), mats should be designed by the conventional rigid method if the spacing of columns in a strip is less than 1.75>b. If the spacing of columns is larger than 1.75>b, the approximate flexible method may be used. To perform the analysis for the structural design of a flexible mat, one must know the principles involved in evaluating the coefficient of subgrade reaction, k. Before proceeding with the discussion of the approximate flexible design method, let us discuss this coefficient in more detail. If a foundation of width B (see Figure 6.12) is subjected to a load per unit area of q, it will undergo a settlement D. The coefficient of subgrade modulus can be defined as k5 q D (6.41) B q Figure 6.12 Definition of coefficient of subgrade reaction, k 311 6.8 Structural Design of Mat Foundations The unit of k is kN/m3. The value of the coefficient of subgrade reaction is not a constant for a given soil, but rather depends on several factors, such as the length L and width B of the foundation and also the depth of embedment of the foundation. A comprehensive study by Terzaghi (1955) of the parameters affecting the coefficient of subgrade reaction indicated that the value of the coefficient decreases with the width of the foundation. In the field, load tests can be carried out by means of square plates measuring 0.3 m 3 0.3 m, and values of k can be calculated. The value of k can be related to large foundations measuring B 3 B in the following ways: Foundations on Sandy Soils For foundations on sandy soils, k 5 k0.3 ¢ B 1 0.3 2 ≤ 2B (6.42) where k0.3 and k 5 coefficients of subgrade reaction of foundations measuring 0.3 m 3 0.3 m and B (m) 3 B (m), respectively (unit is kN>m3 ). Foundations on Clays For foundations on clays, k(kN>m3 ) 5 k0.3 (kN>m3 ) B 0.3 (m) R B(m) (6.43) The definitions of k and k0.3 in Eq. (6.43) are the same as in Eq. (6.42). For rectangular foundations having dimensions of B 3 L (for similar soil and q), k(B3B) ¢ 1 1 0.5 k5 1.5 B ≤ L (6.44) where k 5 coefficient of subgrade modulus of the rectangular foundation (L 3 B) k(B3B) 5 coefficient of subgrade modulus of a square foundation having dimension of B 3 B 312 Chapter 6: Mat Foundations Equation (6.44) indicates that the value of k for a very long foundation with a width B is approximately 0.67k(B3B) . The modulus of elasticity of granular soils increases with depth. Because the settlement of a foundation depends on the modulus of elasticity, the value of k increases with the depth of the foundation. Table 6.2 provides typical ranges of values for the coefficient of subgrade reaction, k0.3 (k1 ), for sandy and clayey soils. For long beams, Vesic (1961) proposed an equation for estimating subgrade reaction, namely, kr 5 Bk 5 0.65 EsB4 Es É EFIF 1 2 m2s 12 or k 5 0.65 EsB4 Es É EFIF B(1 2 m2s ) 12 where Es 5 B5 EF 5 IF 5 ms 5 modulus of elasticity of soil foundation width modulus of elasticity of foundation material moment of inertia of the cross section of the foundation Poisson’s ratio of soil Table 6.2 Typical Subgrade Reaction Values, k0.3 (k1 ) Soil type Dry or moist sand: Loose Medium Dense Saturated sand: Loose Medium Dense Clay: Stiff Very stiff Hard k0.3(k1) MN , m3 8–25 25–125 125–375 10–15 35–40 130–150 10–25 25–50 .50 (6.45) 6.8 Structural Design of Mat Foundations 313 For most practical purposes, Eq. (6.46) can be approximated as k5 Es B(1 2 m2s ) (6.46) Now that we have discussed the coefficient of subgrade reaction, we will proceed with the discussion of the approximate flexible method of designing mat foundations. This method, as proposed by the American Concrete Institute Committee 336 (1988), is described step by step. The use of the design procedure, which is based primarily on the theory of plates, allows the effects (i.e., moment, shear, and deflection) of a concentrated column load in the area surrounding it to be evaluated. If the zones of influence of two or more columns overlap, superposition can be employed to obtain the net moment, shear, and deflection at any point. The method is as follows: Step 1. Assume a thickness h for the mat, according to Step 6 of the conventional rigid method. (Note: h is the total thickness of the mat.) Step 2. Determine the flexural ridigity R of the mat as given by the formula R5 EFh3 12(1 2 m2F ) (6.47) where EF 5 modulus of elasticity of foundation material mF 5 Poisson’s ratio of foundation material Step 3. Determine the radius of effective stiffness—that is, Lr 5 R Åk 4 (6.48) where k 5 coefficient of subgrade reaction. The zone of influence of any column load will be on the order of 3 to 4 Lr. Step 4. Determine the moment (in polar coordinates at a point) caused by a column load (see Figure 6.13a). The formulas to use are Mr 5 radial moment 5 2 Q (1 2 mF )A 2 A 2 S r 4C 1 Lr (6.49) and Mt 5 tangential moment 5 2 Q (1 2 mF )A 2 mFA 1 1 C S r 4 Lr (6.50) 314 Chapter 6: Mat Foundations 6 5 y My Mr 4 Mt Mx r r 3 L ␣ A2 x A4 2 A1 (a) A3 1 0 –0.4 –0.3 –0.2 –0.1 0 0.1 A1, A2, A3, A4 (b) 0.2 0.3 0.4 Figure 6.13 Approximate flexible method of mat design where r 5 radial distance from the column load Q 5 column load A 1 , A 2 5 functions of r>Lr The variations of A 1 and A 2 with r>Lr are shown in Figure 6.13b. (For details see Hetenyi, 1946.) In the Cartesian coordinate system (see Figure 6.13a), Mx 5 Mt sin2 a 1 Mr cos2 a (6.51) My 5 Mt cos2 a 1 Mr sin2 a (6.52) and Step 5. For the unit width of the mat, determine the shear force V caused by a column load: Q V5 A3 (6.53) 4Lr The variation of A 3 with r>Lr is shown in Figure 6.13b. Step 6. If the edge of the mat is located in the zone of influence of a column, determine the moment and shear along the edge. (Assume that the mat is continuous.) Moment and shear opposite in sign to those determined are applied at the edges to satisfy the known conditions. Step 7. The deflection at any point is given by d5 QLr2 A 4R 4 The variation of A 4 is presented in Figure 6.13b. (6.54) 6.8 Structural Design of Mat Foundations 315 Example 6.5 The plan of a mat foundation is shown in Figure 6.14. Calculate the soil pressure at points A, B, C, D, E, and F. (Note: All column sections are planned to be 0.5 m ⫻ 0.5 m.) y y A G B I C 0.25 m 400 kN 500 kN 450 kN 7m 1500 kN 1500 kN 1200 kN 4.25 m 8m 4.25 m 7m x 1500 kN 1500 kN 1200 kN 7m 400 kN 500 kN 350 kN 0.25 m F H E 8m J D x 8m 0.25 m 0.25 m Figure 6.14 Plan of a mat foundation Solution Eq. (6.23): q 5 My x Q Mx y 6 6 A Iy Ix A ⫽ (16.5)(21.5) ⫽ 354.75 m2 1 1 Ix 5 BL3 5 (16.5) (21.5) 3 5 13,665 m4 12 12 1 1 3 Iy 5 LB 5 (21.5) (16.5) 3 5 8,050 m4 12 12 Q ⫽ 350 ⫹ (2)(400) ⫹ 450 ⫹ (2)(500) ⫹ (2)(1200) ⫹ (4)(1500) ⫽ 11,000 kN B My 5 Qex; ex 5 x9 2 2 316 Chapter 6: Mat Foundations Q1x1r 1 Q2x2r 1 Q3x3r 1 c Q (8.25) (500 1 1500 1 1500 1 500) 1 5 C 1 (16.25) (350 1 1200 1 1200 1 450) S 5 7.814 m 11,000 1 (0.25) (400 1 1500 1 1500 1 400) B ex 5 x9 2 5 7.814 2 8.25 5 20.435 m < 20.44 m 2 Hence, the resultant line of action is located to the left of the center of the mat. So My ⫽ (11,000)(0.44) ⫽ 4840 kN-m. Similarly L Mx 5 Qey; ey 5 yr 2 2 Q1y1r 1 Q2y2r 1 Q3y3r 1 c yr 5 Q 1 (0.25) (400 1 500 1 350) 1 (7.25) (1500 1 1500 1 1200) 5 c d 11,000 1(14.25) (1500 1 1500 1 1200) 1 (21.25) (400 1 500 1 450) 5 10.85 m L ey 5 y9 2 5 10.85 2 10.75 5 0.1m 2 The location of the line of action of the resultant column loads is shown in Figure 6.15. Mx ⫽ (11,000)(0.1) ⫽ 1100 kN-m. So 11,000 1100y 4840x q5 6 6 5 31.0 6 0.6x 6 0.08y( kN>m2 ) 354.75 8050 13,665 xr 5 y y A G B I C 0.25 m 400 kN 500 kN 450 kN 7m 1500 kN 4.25 m 0.1 m 1500 kN 1500 kN 1200 kN 8m 4.25 m 0.44 m 7m x 1200 kN 1500 kN 7m 400 kN 500 kN 350 kN 0.25 m F H 8m 0.25 m Figure 6.15 E J 8m D 0.25 m x 6.8 Structural Design of Mat Foundations Therefore, At A: q ⫽ 31.0 ⫹ (0.6)(8.25) ⫹ (0.08)(10.75) ⫽ 36.81 kN/m2 At B: q ⫽ 31.0 ⫹ (0.6)(0) ⫹ (0.08)(10.75) ⫽ 31.86 kN/m2 At C: q ⫽ 31.0 ⫺ (0.6)(8.25) ⫹ (0.08)(10.75) ⫽ 26.91 kN/m2 At D: q ⫽ 31.0 ⫺ (0.6)(8.25) ⫺ (0.08)(10.75) ⫽ 25.19 kN/m2 At E: q ⫽ 31.0 ⫹ (0.6)(0) ⫺ (0.08)(10.75) ⫽ 30.14 kN/m2 At F: q ⫽ 31.0 ⫹ (0.6)(8.25) ⫺ (0.08)(10.75) ⫽ 35.09 kN/m2 317 ■ Example 6.6 Divide the mat shown in Figure 6.14 into three strips, such as AGHF (B1 ⫽ 4.25 m), GIJH (B1 ⫽ 8 m), and ICDJ (B1 ⫽ 4.25 m). Use the result of Example 6.5, and determine the reinforcement requirements in the y direction. Here, fc⬘ ⫽ 20.7 MN/m2, fy ⫽ 413.7 MN/m2, and the load factor is 1.7. Solution Determination of Shear and Moment Diagrams for Strips: Strip AGHF: Average soil pressure 5 qav 5 q(at A) 1 q(at F) 5 36.81 1 35.09 5 35.95 kN>m2 2 Total soil reaction ⫽ qav B1L ⫽ (35.95)(4.25)(21.50) ⫽ 3285 kN load due to soil reaction 1 column loads Average load 5 2 3285 1 3800 5 5 3542.5 kN 2 So, modified average soil pressure, qav(modified) 5 qav a 3542.5 3542.5 b 5 (35.95) a b 5 38.768 kN>m2 3285 3285 The column loads can be modified in a similar manner by multiplying factor F5 3542.5 5 0.9322 3800 Figure 6.16 shows the loading on the strip and corresponding shear and moment diagrams. Note that the column loads shown in this figure have been multiplied by F ⫽ 0.9322. Also the load per unit length of the beam is equal to B1qav(modified) ⫽ (4.25)(38.768) ⫽ 164.76 kN/m. Strip GIJH: In a similar manner, q(at B) 1 q(at E) 31.86 1 30.14 5 5 31.0 kN>m2 2 2 Total soil reaction ⫽ (31)(8)(21.5) ⫽ 5332 kN Total column load ⫽ 4000 kN qav 5 318 Chapter 6: Mat Foundations 372.89 kN 0.25 m 1398.36 kN 1398.36 kN 7m 7m 372.89 kN 0.25 m 7m F A 164.76 kN/m 821.65 576.7 331.7 41.19 Shear (unit: kN) 41.19 381.7 576.70 821.65 1727.57 1127.57 5.15 5.15 718.35 Moment (unit: kN-m) 326.55 326.55 Figure 6.16 Load, shear, and moment diagrams for strip AGHF Average load 5 5332 1 4000 5 4666 kN 2 q av(modified) 5 (31.0) a F5 4666 b 5 27.12 kN>m2 5332 4666 5 1.1665 4000 The load, shear, and moment diagrams are shown in Figure 6.17. Strip ICDJ: Figure 6.18 shows the load, shear, and moment diagrams for this strip. Determination of the Thickness of the Mat For this problem, the critical section for diagonal tension shear will be at the column carrying 1500 kN of load at the edge of the mat [Figure 6.19]. So bo 5 a0.5 1 d d b 1 a0.5 1 b 1 (0.5 1 d) 5 1.5 1 2d 2 2 U 5 (bod) S (f) (0.34)"fcr T or U ⫽ (1.7)(1500) ⫽ 2550 kN ⫽ 2.55 MN 2.55 5 (1.5 1 2d) (d) S(0.85) (0.34) !20.7 T (1.5 ⫹ 2d)(d) ⫽ 1.94; d ⫽ 0.68 m 6.8 Structural Design of Mat Foundations 583.25 kN 0.25 m 1749.75 kN 1749.75 kN 7m 7m 583.25 kN 0.25 m 7m E B 217 kN/m 990.17 759.58 529 54.256 Shear (unit: kN) 54.256 529 759.58 990.17 1620.89 1620.89 291.62 6.75 6.78 Moment (unit: kN-m) 637.94 637.94 Figure 6.17 Load, shear, and moment diagrams for strip GIJH 1046.44 kN 1046.44 kN 392.4 kN 0.25 m 7m 7m 305.2 kN 0.25 m 7m D C 129.8 kN/m 548.65 410.81 272.97 32.45 Shear (unit: kN) 32.3 359.95 497.79 635.63 664.56 360.15 4.06 4.06 Moment (unit: kN-m) 289.95 495.0 1196.19 Figure 6.18 Load, shear, and moment diagrams for strip ICDJ 319 320 Chapter 6: Mat Foundations 1500 kN Column load Edge of mat 0.5 + d 0.5 + d/2 Figure 6.19 Critical perimeter column Assuming a minimum cover of 76 mm over the steel reinforcement and also assuming that the steel bars to be used are 25 mm in diameter, the total thickness of the slab is h ⫽ 0.68 ⫹ 0.076 ⫹ 0.025 ⫽ 0.781 m ⬇ 0.8 m The thickness of this mat will satisfy the wide beam shear condition across the three strips under consideration. Determination of Reinforcement From the moment diagram shown in Figures 6.16, 6.17, and 6.18, it can be seen that the maximum positive moment is located in strip AGHF, and its magnitude is M9 5 1727.57 1727.57 5 5 406.5 kN-m>m B1 4.25 Similarly, the maximum negative moment is located in strip ICDJ and its magnitude is Mr 5 1196.19 1196.19 5 5 281.5 kN-m>m B1 4.25 a From Eq. (6.33): Mu 5 (M r ) ( load factor) 5 fA s fy ad 2 b . 2 a For the positive moment, Mu 5 (406.5)(1.7) 5 (f)(As ) (413.7 3 1000) a0.68 2 b 2 ⫽ 0.9. Also, from Eq. (6.34), As fy (As )(413.7) 5 23.51As; or As 5 0.0425a (0.85) (20.7) (1) a 691.05 5 (0.9)(0.0425a) (413,700) a0.68 2 b; or a < 0.0645 2 a5 0.85 f9c b 5 So, As ⫽ (0.0425)(0.0645) ⫽ 0.00274 m2/m ⫽ 2740 mm2/m. 6.8 Structural Design of Mat Foundations 321 Use 25-mm diameter bars at 175 mm center-to-center: cA s provided 5 (491) a 1000 b 5 2805.7 mm2>md 175 Similarly, for negative reinforcement, a Mu 5 (281.5) (1.7) 5 (f)(As )(413.7 3 1000) a0.68 2 b 2 ⫽ 0.9. As ⫽ 0.0425a So a 478.55 5 (0.9) (0.0425a) (413.7 3 1000) a0.68 2 b; or a < 0.045 2 So, As ⫽ (0.045)(0.0425) ⫽ 0.001913 m2/m ⫽ 1913 mm2/m. Use 25-mm diameter bars at 255 mm center-to-center: [As provided ⫽ 1925 mm2] Because negative moment occurs at midbay of strip ICDJ, reinforcement should be provided. This moment is M9 5 289.95 5 68.22 kN- m>m 4.25 Hence, a Mu 5 (68.22)(1.7) 5 (0.9)(0.0425a)(413.7 3 1000) a0.68 2 b; 2 or a < 0.0108 As ⫽ (0.0108)(0.0425) ⫽ 0.000459 m2/m ⫽ 459 mm2/m Provide 16-mm diameter bars at 400 mm center-to-center: [As provided ⫽ 502 mm2] For general arrangement of the reinforcement see Figure 6.20. Top steel Bottom steel Additional top steel in strip ICDJ Top steel Figure 6.20 General arrangement of reinforcement ■ 322 Chapter 6: Mat Foundations Problems 6.1 Determine the net ultimate bearing capacity of mat foundations with the following characteristics: cu 5 120 kN>m2, f 5 0, B 5 8 m, L 5 18 m, Df 5 3 m 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 Following are the results of a standard penetration test in the field (sandy soil): Depth (m) Field value of N60 1.5 3.0 4.5 6.0 7.5 9.0 10.5 9 12 11 7 13 11 13 Estimate the net allowable bearing capacity of a mat foundation 6.5 m 3 5 m in plan. Here, Df 5 1.5 m and allowable settlement 5 50 mm. Assume that the unit weight of soil, g 5 16.5 kN>m3. Repeat Problem 6.2 for an allowable settlement of 30 mm. A mat foundation on a saturated clay soil has dimensions of 20 m 3 20 m. Given: dead and live load 5 48 MN, cu 5 30 kN>m2, and gclay 5 18.5 kN>m3. a. Find the depth, Df, of the mat for a fully compensated foundation. b. What will be the depth of the mat (Df ) for a factor of safety of 2 against bearing capacity failure? Repeat Problem 6.4 part b for cu 5 20 kN>m2 . A mat foundation is shown in Figure P6.6. The design considerations are L 5 12 m, B 5 10 m, Df 5 2.2 m, Q 5 30 MN, x1 5 2 m, x2 5 2 m, x3 5 5.2 m, and preconsolidation pressure scr < 105 kN>m 2. Calculate the consolidation settlement under the center of the mat. For the mat foundation in Problem 6.6, estimate the consolidation settlement under the corner of the mat. From the plate load test (plate dimensions 0.3 m 3 0.3 m) in the field, the coefficient of subgrade reaction of a sandy soil is determined to be 14,900 kN>m3. What will be the value of the coefficient of subgrade reaction on the same soil for a foundation with dimensions of 7.5 m 3 7.5 m? Refer to Problem 6.18. If the full-sized foundation had dimensions of 21.3 m 3 9.1 m, what will be the value of the coefficient of subgrade reaction? The subgrade reaction of a sandy soil obtained from the plate load test (plate dimensions 1 m 3 0.7 m ) is 18 MN>m3. What will be the value of k on the same soil for a foundation measuring 5 m 3 3.5 m ? References 323 Size of mat B L Sand ␥ 16.0 kN/m3 Df Q x1 z x2 x3 Groundwater table Sand ␥ sat 18.0 kN/m3 Clay ␥sat 17.5 kN/m3 eo 0.88 Cc 0.38 Cs 0.1 Figure P6.6 References AMERICAN CONCRETE INSTITUTE (1995). ACI Standard Building Code Requirements for Reinforced Concrete, ACI 318–95, Farmington Hills, MI. AMERICAN CONCRETE INSTITUTE COMMITTEE 336 (1988). “Suggested Design Procedures for Combined Footings and Mats,” Journal of the American Concrete Institute, Vol. 63, No. 10, pp. 1041–1077. HETENYI, M. (1946). Beams of Elastic Foundations, University of Michigan Press, Ann Arbor, MI. MEYERHOF, G. G. (1965). “Shallow Foundations,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 91, No. SM2, pp. 21–31. RIOS, L., and SILVA, F. P. (1948). “Foundations in Downtown São Paulo (Brazil),” Proceedings, Second International Conference on Soil Mechanics and Foundation Engineering, Rotterdam, Vol. 4, p. 69. SCHULTZE, E. (1962). “Probleme bei der Auswertung von Setzungsmessungen,” Proceedings, Baugrundtagung, Essen, Germany, p. 343. TERZAGHI, K. (1955). “Evaluation of the Coefficient of Subgrade Reactions,” Geotechnique, Institute of Engineers, London, Vol. 5, No. 4, pp. 197–226. VARGAS, M. (1948). “Building Settlement Observations in São Paulo,” Proceedings Second International Conference on Soil Mechanics and Foundation Engineering, Rotterdam, Vol. 4, p. 13. VARGAS, M. (1961). “Foundations of Tall Buildings on Sand in São Paulo (Brazil),” Proceedings, Fifth International Conference on Soil Mechanics and Foundation Engineering, Paris, Vol. 1, p. 841. VESIC, A. S. (1961). “Bending of Beams Resting on Isotropic Solid,” Journal of the Engineering Mechanics Division, American Society of Civil Engineers, Vol. 87, No. EM2, pp. 35–53. 7 7.1 Lateral Earth Pressure Introduction Vertical or near-vertical slopes of soil are supported by retaining walls, cantilever sheet-pile walls, sheet-pile bulkheads, braced cuts, and other, similar structures. The proper design of those structures requires an estimation of lateral earth pressure, which is a function of several factors, such as (a) the type and amount of wall movement, (b) the shear strength parameters of the soil, (c) the unit weight of the soil, and (d) the drainage conditions in the backfill. Figure 7.1 shows a retaining wall of height H. For similar types of backfill, a. The wall may be restrained from moving (Figure 7.1a). The lateral earth pressure on the wall at any depth is called the at-rest earth pressure. b. The wall may tilt away from the soil that is retained (Figure 7.1b). With sufficient wall tilt, a triangular soil wedge behind the wall will fail. The lateral pressure for this condition is referred to as active earth pressure. c. The wall may be pushed into the soil that is retained (Figure 7.1c). With sufficient wall movement, a soil wedge will fail. The lateral pressure for this condition is referred to as passive earth pressure. Figure 7.2 shows the nature of variation of the lateral pressure, shr , at a certain depth of the wall with the magnitude of wall movement. + H – H h (at-rest) h (active) h (passive) Soil failure wedge Height = H Height = H (a) 324 Soil failure wedge Height = H (b) Figure 7.1 Nature of lateral earth pressure on a retaining wall (c) 7.2 Lateral Earth Pressure at Rest 325 h h (passive) ( ) H H p ( ) H H ⬇ 0.01 for loose sand to 0.05 for soft clay ( ) H H H H h (active) p a ⬇ 0.001 for loose sand to 0.04 for soft clay h (at rest) ( ) H H a H H Figure 7.2 Nature of variation of lateral earth pressure at a certain depth In the sections that follow, we will discuss various relationships to determine the at-rest, active, and passive pressures on a retaining wall. It is assumed that the reader has studied lateral earth pressure in the past, so this chapter will serve as a review. 7.2 Lateral Earth Pressure at Rest Consider a vertical wall of height H, as shown in Figure 7.3, retaining a soil having a unit weight of g. A uniformly distributed load, q>unit area, is also applied at the ground surface. The shear strength of the soil is s 5 cr 1 sr tan fr where cr 5 cohesion fr 5 effective angle of friction sr 5 effective normal stress q Ko q o c z H 1 2 P1 Po h P2 H/2 H/3 Ko (q + H) (a) Figure 7.3 At-rest earth pressure (b) z 326 Chapter 7: Lateral Earth Pressure At any depth z below the ground surface, the vertical subsurface stress is (7.1) sor 5 q 1 gz If the wall is at rest and is not allowed to move at all, either away from the soil mass or into the soil mass (i.e., there is zero horizontal strain), the lateral pressure at a depth z is (7.2) sh 5 Kosor 1 u where u 5 pore water pressure Ko 5 coefficient of at-rest earth pressure For normally consolidated soil, the relation for Ko (Jaky, 1944) is Ko < 1 2 sin fr (7.3) Equation (7.3) is an empirical approximation. For overconsolidated soil, the at-rest earth pressure coefficient may be expressed as (Mayne and Kulhawy, 1982) Ko 5 (1 2 sin fr)OCRsin fr (7.4) where OCR 5 overconsolidation ratio. With a properly selected value of the at-rest earth pressure coefficient, Eq. (7.2) can be used to determine the variation of lateral earth pressure with depth z. Figure 7.3b shows the variation of shr with depth for the wall depicted in Figure 7.3a. Note that if the surcharge q 5 0 and the pore water pressure u 5 0, the pressure diagram will be a triangle. The total force, Po , per unit length of the wall given in Figure 7.3a can now be obtained from the area of the pressure diagram given in Figure 7.3b and is Po 5 P1 1 P2 5 qKoH 1 12gH 2Ko (7.5) where P1 5 area of rectangle 1 P2 5 area of triangle 2 The location of the line of action of the resultant force, Po , can be obtained by taking the moment about the bottom of the wall. Thus, P1 ¢ z5 H H ≤ 1 P2 ¢ ≤ 2 3 Po (7.6) If the water table is located at a depth z , H, the at-rest pressure diagram shown in Figure 7.3b will have to be somewhat modified, as shown in Figure 7.4. If the effective unit weight of soil below the water table equals gr (i.e., gsat 2 gw ), then at z 5 0, at z 5 H1 , shr 5 Kosor 5 Koq shr 5 Kosor 5 Ko (q 1 gH1 ) and at z 5 H2 , shr 5 Kosor 5 Ko (q 1 gH1 1 grH2 ) 7.2 Lateral Earth Pressure at Rest 327 q Ko q c H1 1 z 2 Water table Ko (q + H1 ) H 3 sat c H2 h u 5 4 Ko (q + H1 + H2 ) (a) w H2 (b) Figure 7.4 At-rest earth pressure with water table located at a depth z , H Note that in the preceding equations, sor and shr are effective vertical and horizontal pressures, respectively. Determining the total pressure distribution on the wall requires adding the hydrostatic pressure u, which is zero from z 5 0 to z 5 H1 and is H2gw at z 5 H2 . The variation of shr and u with depth is shown in Figure 7.4b. Hence, the total force per unit length of the wall can be determined from the area of the pressure diagram. Specifically, Po 5 A 1 1 A 2 1 A 3 1 A 4 1 A 5 where A 5 area of the pressure diagram. So, Po 5 KoqH1 1 12KogH 21 1 Ko (q 1 gH1 )H2 1 12KogrH 22 1 12gwH 22 (7.7) Example 7.1 For the retaining wall shown in Figure 7.5(a), determine the lateral earth force at rest per unit length of the wall. Also determine the location of the resultant force. Assume OCR ⫽ 1. Solution Ko ⫽ 1 ⫺ sin ⬘ ⫽ 1 ⫺ sin 30° ⫽ 0.5 At z ⫽ 0, o⬘ ⫽ 0; h⬘ ⫽ 0 At z ⫽ 2.5 m, o⬘ ⫽ (16.5)(2.5) ⫽ 41.25 kN/m2; h⬘ ⫽ Koo⬘ ⫽ (0.5)(41.25) ⫽ 20.63 kN/m2 At z ⫽ 5 m, o⬘ ⫽ (16.5)(2.5) ⫹ (19.3 ⫺ 9.81)2.5 ⫽ 64.98 kN/m2; h⬘ ⫽ Koo⬘ ⫽ (0.5)(64.98) ⫽ 32.49 kN/m2 328 Chapter 7: Lateral Earth Pressure = 16.5 kN/m3 = 30 c = 0 Ground water table 2.5 m z sat = 19.3 kN/m3 = 30 c = 0 2.5 m 1 20.63 kN/m2 2 h 3 O (a) 32.49 kN/m2 u 4 24.53 kN/m2 (b) Figure 7.5 The hydrostatic pressure distribution is as follows: From z ⫽ 0 to z ⫽ 2.5 m, u ⫽ 0. At z ⫽ 5 m, u ⫽ ␥w(2.5) ⫽ (9.81)(2.5) ⫽ 24.53 kN/m2. The pressure distribution for the wall is shown in Figure 7.5b. The total force per unit length of the wall can be determined from the area of the pressure diagram, or Po 5 Area 1 1 Area 2 1 Area 3 1 Area 4 5 12 (2.5) (20.63) 1 (2.5) (20.63) 1 12 (2.5) (32.49 2 20.63) 1 12 (2.5) (24.53) 5 122.85 kN>m The location of the center of pressure measured from the bottom of the wall (point O) ⫽ ( z5 Area 1)a2.51 2.5 3 b1( Area 2)a 2.5 2 b1( Area 31 Area 4)a 2.5 3 b Po (25.788) (3.33) 1 (51.575) (1.25) 1 (14.825 1 30.663) (0.833) 122.85 85.87 1 64.47 1 37.89 5 5 1.53 m 122.85 5 ■ Active Pressure 7.3 Rankine Active Earth Pressure The lateral earth pressure described in Section 7.2 involves walls that do not yield at all. However, if a wall tends to move away from the soil a distance Dx, as shown in Figure 7.6a, the soil pressure on the wall at any depth will decrease. For a wall that is frictionless, the horizontal stress, shr , at depth z will equal Kosro (5Kogz) when Dx is zero. However, with Dx . 0, shr will be less than Kosor . Wall movement to left 45 /2 x 45 /2 z z o c H h Rotation of wall about this point (a) Shear stress s c ta c b n a a h Koo o Effective normal stress (b) 2c Ka zc H o Ka 2c Ka Pa Eq. (7.12) o Ka 2c Ka (c) Figure 7.6 Rankine active pressure 329 330 Chapter 7: Lateral Earth Pressure The Mohr’s circles corresponding to wall displacements of Dx 5 0 and Dx . 0 are shown as circles a and b, respectively, in Figure 7.6b. If the displacement of the wall, Dx, continues to increase, the corresponding Mohr’s circle eventually will just touch the Mohr–Coulomb failure envelope defined by the equation s 5 cr 1 sr tan fr This circle, marked c in the figure, represents the failure condition in the soil mass; the horizontal stress then equals sar , referred to as the Rankine active pressure. The slip lines (failure planes) in the soil mass will then make angles of 6(45 1 fr>2) with the horizontal, as shown in Figure 7.6a. Equation (1.87) relates the principal stresses for a Mohr’s circle that touches the Mohr–Coulomb failure envelope: s1r 5 s3r tan2 ¢ 45 1 fr fr ≤ 1 2cr tan ¢ 45 1 ≤ 2 2 For the Mohr’s circle c in Figure 7.6b, Major principal stress, s1r 5 sor and Minor principal stress, s3r 5 sar Thus, fr fr ≤ 1 2cr tan ¢45 1 ≤ 2 2 sor 2cr 2 sar 5 fr fr 2 tan ¢45 1 ≤ tan ¢45 1 ≤ 2 2 sor 5 sar tan2 ¢ 45 1 or sar 5 sor tan2 ¢ 45 2 fr fr ≤ 2 2cr tan ¢45 2 ≤ 2 2 5 sor Ka 2 2cr"Ka (7.8) where Ka 5 tan2 (45 2 fr>2) 5 Rankine active-pressure coefficient. The variation of the active pressure with depth for the wall shown in Figure 7.6a is given in Figure 7.6c. Note that sor 5 0 at z 5 0 and sor 5 gH at z 5 H. The pressure distribution shows that at z 5 0 the active pressure equals 22cr"Ka , indicating a tensile stress that decreases with depth and becomes zero at a depth z 5 zc , or gzcKa 2 2cr"Ka 5 0 7.3 Rankine Active Earth Pressure 331 and zc 5 2cr (7.9) g"Ka The depth zc is usually referred to as the depth of tensile crack, because the tensile stress in the soil will eventually cause a crack along the soil–wall interface. Thus, the total Rankine active force per unit length of the wall before the tensile crack occurs is H H H Pa 5 3 sar dz 5 3 gzKa dz 2 3 2cr"Ka dz 0 0 0 5 12gH 2Ka 2 2crH"Ka (7.10) After the tensile crack appears, the force per unit length on the wall will be caused only by the pressure distribution between depths z 5 zc and z 5 H, as shown by the hatched area in Figure 7.6c. This force may be expressed as Pa 5 12 (H 2 zc ) (gHKa 2 2cr"Ka ) (7.11) 1 2cr ≤ agHKa 2 2cr"Ka b Pa 5 ¢ H 2 2 g"Ka (7.12) or However, it is important to realize that the active earth pressure condition will be reached only if the wall is allowed to “yield” sufficiently. The necessary amount of outward displacement of the wall is about 0.001H to 0.004H for granular soil backfills and about 0.01H to 0.04H for cohesive soil backfills. Note further that if the total stress shear strength parameters (c, f) were used, an equation similar to Eq. (7.8) could have been derived, namely, sa 5 so tan2 ¢ 45 2 f f ≤ 2 2c tan ¢45 2 ≤ 2 2 Example 7.2 A 6-m-high retaining wall is to support a soil with unit weight ␥ ⫽ 17.4 kN/m3, soil friction angle ⬘ ⫽ 26°, and cohesion c⬘ ⫽ 14.36 kN/m2. Determine the Rankine active force per unit length of the wall both before and after the tensile crack occurs, and determine the line of action of the resultant in both cases. Solution For ⬘ ⫽ 26°, Ka 5 tan2 a45 2 !Ka 5 0.625 fr b 5 tan2 (45 2 13) 5 0.39 2 sar 5 gHKa 2 2cr !Ka 332 Chapter 7: Lateral Earth Pressure From Figure 7.6c, at z ⫽ 0, sar 5 22cr !Ka 5 22(14.36) (0.625) 5 217.95 kN>m2 and at z ⫽ 6 m, a⬘ ⫽ (17.4)(6)(0.39) ⫺ 2(14.36)(0.625) ⫽ 40.72 ⫺ 17.95 ⫽ 22.77 kN/m2 Active Force before the Tensile Crack Appeared: Eq. (7.10) Pa 5 12 gH 2Ka 2 2crH!Ka 5 12 (6) (40.72) 2 (6) (17.95) 5 122.16 2 107.7 5 14.46 kN>m The line of action of the resultant can be determined by taking the moment of the area of the pressure diagrams about the bottom of the wall, or 6 6 Paz 5 (122.16) a b 2 (107.7) a b 3 2 Thus, z5 244.32 2 323.1 5 25.45 m. 14.46 Active Force after the Tensile Crack Appeared: Eq. (7.9) zc 5 2(14.36) 2cr 5 5 2.64 m (17.4) (0.625) g !Ka Using Eq. (7.11) gives Pa 5 12 (H 2 zc ) (gHKa 2 2c9 !Ka ) 5 12 (6 2 2.64) (22.77) 5 38.25 kN>m Figure 7.6c indicates that the force Pa ⫽ 38.25 kN/m is the area of the hatched triangle. Hence, the line of action of the resultant will be located at a height –z ⫽ (H ⫺ zc)/3 above the bottom of the wall, or z5 6 2 2.64 5 1.12 m 3 ■ Example 7.3 Assume that the retaining wall shown in Figure 7.7a can yield sufficiently to develop an active state. Determine the Rankine active force per unit length of the wall and the location of the resultant line of action. Solution If the cohesion, c⬘, is zero, then a⬘ ⫽ o⬘Ka 7.3 Rankine Active Earth Pressure 333 For the top layer of soil, 1⬘ ⫽ 30°, so Ka(1) 5 tan2 a45 2 fr1 1 b 5 tan2 (45 2 15) 5 2 3 Similarly, for the bottom layer of soil, 2⬘ ⫽ 36°, and it follows that Ka(2) 5 tan2 a45 2 36 b 5 0.26 2 The following table shows the calculation of a⬘ and u at various depths below the ground surface. Depth, z (m) o⬘ (lb/ft2) Ka a⬘ ⫽ Kao⬘ (lb/ft2) u (lb/ft2) 0 3.05⫺ 3.05⫹ 6.1 0 (16)(3.05) ⫽ 48.8 48.8 (16)(3.05) ⫹ (19 ⫺ 9.81)(3.05) ⫽ 76.83 1/3 1/3 0.26 0.26 0 16.27 12.69 19.98 0 0 0 (9.81)(3.05) ⫽ 29.92 The pressure distribution diagram is plotted in Figure 7.7b. The force per unit length is Pa ⫽ area 1 ⫹ area 2 ⫹ area 3 ⫹ area 4 5 12 (3.05)(16.27) 1 (12.69)(3.05) 1 12 (19.98 2 12.69)(3.05) 1 12 (29.92)(3.05) ⫽ 24.81 ⫹ 38.70 ⫹ 11.12 ⫹ 45.63 ⫽ 120.26 kN/m = 16 kN/m3 ′1 = 30° c′1 = 0 3.05 m z Water table sat = 19 kN/m3 ′2 = 36° c′2 = 0 3.05 m 1 12.69 kN/m2 16.27 kN/m2 2 3 4 19.98 kN/m2 0 (a) Figure 7.7 Rankine active force behind a retaining wall 29.92 kN/m2 (b) ■ 334 Chapter 7: Lateral Earth Pressure The distance of the line of action of the resultant force from the bottom of the wall can be determined by taking the moments about the bottom of the wall (point O in Figure 7.7a) and is 3.05 3.05 3.05 (24.81) a3.05 1 b 1 (38.7) a b 1 (11.12 1 45.63) a b 3 2 3 z5 5 1.81 m ■ 120.26 7.4 A Generalized Case for Rankine Active Pressure In Section 7.3, the relationship was developed for Rankine active pressure for a retaining wall with a vertical back and a horizontal backfill. That can be extended to general cases of frictionless walls with inclined backs and inclined backfills. Some of these cases will be discussed in this section. Granular Backfill Figure 7.8 shows a retaining wall whose back is inclined at an angle u with the vertical. The granular backfill is inclined at an angle a with the horizontal. For a Rankine active case, the lateral earth pressure (sar ) at a depth z can be given as (Chu, 1991), sar 5 where ca 5 sin21 ¢ g z cos a"1 1 sin2 fr 2 2 sin fr cos ca cos a 1 "sin2 fr 2 sin2 a sin a ≤ 2 a 1 2u. sin fr (7.13) (7.14) z a H Frictionless wall Figure 7.8 General case for a retaining wall with granular backfill 7.4 A Generalized Case for Rankine Active Pressure 335 The pressure sar will be inclined at an angle br with the plane drawn at right angle to the backface of the wall, and br 5 tan21 ¢ sin fr sin ca ≤ 1 2 sin fr cos ca (7.15) The active force Pa for unit length of the wall then can be calculated as Pa 5 1 gH 2Ka 2 (7.16) where Ka 5 cos(a 2 u)"1 1 sin2 fr 2 2 sin fr cos ca cos2 u Q cos a 1 "sin2 fr 2 sin2 a R 5 Rankine active earth-pressure coefficient for generalized case (7.17) The location and direction of the resultant force Pa is shown in Figure 7.9. Also shown in this figure is the failure wedge, ABC. Note that BC will be inclined at an angle h. Or h5 fr p a 1 sin a 1 1 2 sin21 ¢ ≤ 4 2 2 2 sin fr (7.18) C A Failure wedge Pa H H/3 B 4 1 sin sin1 2 2 2 sin ( ) Figure 7.9 Location and direction of Rankine active force 336 Chapter 7: Lateral Earth Pressure Granular Backfill with Vertical Back Face As a special case, for a vertical backface of a wall (that is, u 5 0), as shown in Figure 7.10, Eqs. (7.13), (7.16) and (7.17) simplify to the following. If the backfill of a frictionless retaining wall is a granular soil (cr 5 0) and rises at an angle a with respect to the horizontal (see Figure 7.10), the active earth-pressure coefficient may be expressed in the form Ka 5 cos a cos a2"cos2 a2cos2 fr cos a 1 "cos2 a2cos2 fr (7.19) where fr 5 angle of friction of soil. At any depth z, the Rankine active pressure may be expressed as sar 5 gzKa (7.20) Also, the total force per unit length of the wall is Pa 5 12 gH 2Ka (7.21) Note that, in this case, the direction of the resultant force Pa is inclined at an angle a with the horizontal and intersects the wall at a distance H>3 from the base of the wall. Table 7.1 presents the values of Ka (active earth pressure) for various values of a and fr. a z Pa H H/3 Figure 7.10 Notations for active pressure—Eqs. (7.19), (7.20), (7.21) 337 28 0.3610 0.3612 0.3618 0.3627 0.3639 0.3656 0.3676 0.3701 0.3730 0.3764 0.3802 0.3846 0.3896 0.3952 0.4015 0.4086 0.4165 0.4255 0.4357 0.4473 0.4605 0.4758 0.4936 0.5147 0.5404 0.5727 T 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 ␣ (deg) 0.3470 0.3471 0.3476 0.3485 0.3496 0.3512 0.3531 0.3553 0.3580 0.3611 0.3646 0.3686 0.3731 0.3782 0.3839 0.3903 0.3975 0.4056 0.4146 0.4249 0.4365 0.4498 0.4651 0.4829 0.5041 0.5299 29 30 0.3333 0.3335 0.3339 0.3347 0.3358 0.3372 0.3389 0.3410 0.3435 0.3463 0.3495 0.3532 0.3573 0.3620 0.3671 0.3729 0.3794 0.3867 0.3948 0.4039 0.4142 0.4259 0.4392 0.4545 0.4724 0.4936 Table 7.1 Values of Ka [Eq. (7.19)] 0.3201 0.3202 0.3207 0.3214 0.3224 0.3237 0.3253 0.3272 0.3294 0.3320 0.3350 0.3383 0.3421 0.3464 0.3511 0.3564 0.3622 0.3688 0.3761 0.3842 0.3934 0.4037 0.4154 0.4287 0.4440 0.4619 31 0.3073 0.3074 0.3078 0.3084 0.3094 0.3105 0.3120 0.3138 0.3159 0.3182 0.3210 0.3241 0.3275 0.3314 0.3357 0.3405 0.3458 0.3518 0.3584 0.3657 0.3739 0.3830 0.3934 0.4050 0.4183 0.4336 32 0.2948 0.2949 0.2953 0.2959 0.2967 0.2978 0.2992 0.3008 0.3027 0.3049 0.3074 0.3103 0.3134 0.3170 0.3209 0.3253 0.3302 0.3356 0.3415 0.3481 0.3555 0.3637 0.3729 0.3832 0.3948 0.4081 33 0.2827 0.2828 0.2832 0.2837 0.2845 0.2855 0.2868 0.2883 0.2900 0.2921 0.2944 0.2970 0.2999 0.3031 0.3068 0.3108 0.3152 0.3201 0.3255 0.3315 0.3381 0.3455 0.3537 0.3628 0.3731 0.3847 34 ⴕ (deg) S 0.2710 0.2711 0.2714 0.2719 0.2726 0.2736 0.2747 0.2761 0.2778 0.2796 0.2818 0.2841 0.2868 0.2898 0.2931 0.2968 0.3008 0.3053 0.3102 0.3156 0.3216 0.3283 0.3356 0.3438 0.3529 0.3631 35 0.2596 0.2597 0.2600 0.2605 0.2611 0.2620 0.2631 0.2644 0.2659 0.2676 0.2696 0.2718 0.2742 0.2770 0.2800 0.2834 0.2871 0.2911 0.2956 0.3006 0.3060 0.3120 0.3186 0.3259 0.3341 0.3431 36 0.2486 0.2487 0.2489 0.2494 0.2500 0.2508 0.2518 0.2530 0.2544 0.2560 0.2578 0.2598 0.2621 0.2646 0.2674 0.2705 0.2739 0.2776 0.2817 0.2862 0.2911 0.2965 0.3025 0.3091 0.3164 0.3245 37 0.2379 0.2380 0.2382 0.2386 0.2392 0.2399 0.2409 0.2420 0.2432 0.2447 0.2464 0.2482 0.2503 0.2527 0.2552 0.2581 0.2612 0.2646 0.2683 0.2724 0.2769 0.2818 0.2872 0.2932 0.2997 0.3070 38 0.2275 0.2276 0.2278 0.2282 0.2287 0.2294 0.2303 0.2313 0.2325 0.2338 0.2354 0.2371 0.2390 0.2412 0.2435 0.2461 0.2490 0.2521 0.2555 0.2593 0.2634 0.2678 0.2727 0.2781 0.2840 0.2905 39 0.2174 0.2175 0.2177 0.2181 0.2186 0.2192 0.2200 0.2209 0.2220 0.2233 0.2247 0.2263 0.2281 0.2301 0.2322 0.2346 0.2373 0.2401 0.2433 0.2467 0.2504 0.2545 0.2590 0.2638 0.2692 0.2750 40 338 Chapter 7: Lateral Earth Pressure Vertical Backface with c9– f9 Soil Backfill For a retaining wall with a vertical back (u 5 0) and inclined backfill of cr–fr soil (Mazindrani and Ganjali, 1997), sar 5 gzKa 5 gzKar cos a (7.22) where Kar 5 1 d cos2 fr 2 cos2 a 1 2¢ cr ≤ cos fr sin fr gz cr 2 cr 2 B4 cos a(cos a2cos fr) 1 4¢ ≤ cos2 fr 1 8¢ ≤ cos2 a sin fr cos fr R gz gz Å 2 2 t 21 2 (7.23) Some values of Kar are given in Table 7.2. For a problem of this type, the depth of tensile crack is given as zc 5 2cr 1 1 sin fr g Å 1 2 sin fr (7.24) For this case, the active pressure is inclined at an angle a with the horizontal. Table 7.2 Values of Kar c9 gz f9 (deg) a (deg) 0.025 0.05 0.1 0.5 15 0 5 10 15 0 5 10 15 0 5 10 15 0 5 10 15 0.550 0.566 0.621 0.776 0.455 0.465 0.497 0.567 0.374 0.381 0.402 0.443 0.305 0.309 0.323 0.350 0.512 0.525 0.571 0.683 0.420 0.429 0.456 0.514 0.342 0.348 0.366 0.401 0.276 0.280 0.292 0.315 0.435 0.445 0.477 0.546 0.350 0.357 0.377 0.417 0.278 0.283 0.296 0.321 0.218 0.221 0.230 0.246 20.179 20.184 20.186 20.196 20.210 20.212 20.218 20.229 20.231 20.233 20.239 20.250 20.244 20.246 20.252 20.263 20 25 30 7.4 A Generalized Case for Rankine Active Pressure 339 Example 7.4 Refer to the retaining wall in Figure 7.9. The backfill is granular soil. Given: Wall: H 5 10 ft u 5 110° Backfill: a 5 15° fr 5 35° cr 5 0 g 5 110 lb>ft 3 Determine the Rankine active force, Pa, and its location and direction. Solution From Eq. (7.14), ca 5 sin21 ¢ sin a sin 15 ≤ 2 a 1 2u 5 sin21 ¢ ≤ 2 15 1 (2) (10) 5 31.82° sin fr sin 35 From Eq. (7.17), Ka 5 5 cos(a 2 u)"1 1 sin2 fr 2 2 sin fr cos ca cos2 u Q cos a 1 "sin2 fr 2 sin2 a R cos(15 2 10)"1 1 sin2 35 2 (2) (sin 35) (sin 31.82) cos2 10 Q cos 15 1 "sin2 35 2 sin2 15 R 5 0.59 Pa 5 1⁄2 gH 2Ka 5 ( 1⁄2 ) (110) (10) 2 (0.59) 5 3245 lb , ft From Eq. (7.15), br 5 tan21 ¢ sin fr sin ca (sin 35) (sin 31.82) ≤ 5 tan21 B R 5 30.58 1 2 sin fr cos ca 1 2 (sin 35) (cos 31.82) The force Pa will act at a distance of 10>3 5 3.33 ft above the bottom of the wall and will be inclined at an angle of130.5° to the normal drawn to the back face of the wall. ■ Example 7.5 For the retaining wall shown in Figure 7.10, H ⫽ 7.5 m, ␥ ⫽ 18 kN/m3, ⬘ ⫽ 20°, c⬘ ⫽ 13.5 kN/m2, and ␣ ⫽ 10°. Calculate the Rankine active force, Pa, per unit length of the wall and the location of the resultant force after the occurrence of the tensile crack. Solution From Eq. (7.24). zr 5 (2) (13.5) 1 1 sin 20 2c r 1 1 sinfr 5 5 2.14 m g Å 1 2 sinfr 18 Å 1 2 sin 20 340 Chapter 7: Lateral Earth Pressure At z ⫽ 7.5 m, cr 13.5 5 5 0.1 gz (18) (7.5) From Table 7.2, for ⬘ ⫽ 20°, c⬘/␥z ⫽ 0.1, and ␣ ⫽ 10°, the value of Ka⬘ is 0.377, so at z ⫽ 7.5 m, a⬘ ⫽ ␥zKa⬘cos ␣ ⫽ (18)(7.5)(0.377)(cos 10) ⫽ 50.1 kN/m2 After the occurrence of the tensile crack, the pressure distribution on the wall will be as shown in Figure 7.11, so 1 Pa 5 a b (50.1) (7.5 2 2.14) 5 134.3 kN>m 2 and z5 7.5 2 2.14 5 1.79 m 3 ■ 10° 2.14 m 5.36 m Pa z = 1.79 m 10° 7.5 Figure 7.11 Calculation of Rankine active force, c⬘ ⫺ ⬘ soil Coulomb’s Active Earth Pressure The Rankine active earth pressure calculations discussed in the preceding sections were based on the assumption that the wall is frictionless. In 1776, Coulomb proposed a theory for calculating the lateral earth pressure on a retaining wall with granular soil backfill. This theory takes wall friction into consideration. To apply Coulomb’s active earth pressure theory, let us consider a retaining wall with its back face inclined at an angle b with the horizontal, as shown in Figure 7.12a. The backfill is a granular soil that slopes at an angle a with the horizontal. 7.5 Coulomb’s Active Earth Pressure 341 Pa(max) Active force C3 C2 Wall movement C1 away from soil A Pa c = 0 W N H Pa R 1 R S H/3 W (b) 1 B (a) Figure 7.12 Coulomb’s active pressure Also, let dr be the angle of friction between the soil and the wall (i.e., the angle of wall friction). Under active pressure, the wall will move away from the soil mass (to the left in the figure). Coulomb assumed that, in such a case, the failure surface in the soil mass would be a plane (e.g., BC1 , BC2 , c ). So, to find the active force, consider a possible soil failure wedge ABC1 . The forces acting on this wedge (per unit length at right angles to the cross section shown) are as follows: 1. The weight of the wedge, W. 2. The resultant, R, of the normal and resisting shear forces along the surface, BC1 . The force R will be inclined at an angle fr to the normal drawn to BC1 . 3. The active force per unit length of the wall, Pa , which will be inclined at an angle dr to the normal drawn to the back face of the wall. For equilibrium purposes, a force triangle can be drawn, as shown in Figure 7.12b. Note that u1 is the angle that BC1 makes with the horizontal. Because the magnitude of W, as well as the directions of all three forces, are known, the value of Pa can now be determined. Similarly, the active forces of other trial wedges, such as ABC2 , ABC3 , c, can be determined. The maximum value of Pa thus determined is Coulomb’s active force (see top part of Figure 7.12), which may be expressed as Pa 5 12KagH 2 (7.25) 342 Chapter 7: Lateral Earth Pressure where Ka 5 Coulomb’s active earth pressure coefficient 5 sin2 (b 1 fr) (7.26) sin (fr 1 dr)sin (fr2a) 2 sin b sin (b2dr) B1 1 R Å sin (b2dr)sin (a 1 b) 2 and H 5 height of the wall. The values of the active earth pressure coefficient, Ka , for a vertical retaining wall (b 5 90°) with horizontal backfill (a 5 0°) are given in Table 7.3. Note that the line of action of the resultant force (Pa ) will act at a distance H>3 above the base of the wall and will be inclined at an angle dr to the normal drawn to the back of the wall. In the actual design of retaining walls, the value of the wall friction angle dr is assumed to be between fr>2 and 23fr. The active earth pressure coefficients for various values of fr, a, and b with dr 5 12fr and 32fr are respectively given in Tables 7.4 and 7.5. These coefficients are very useful design considerations. If a uniform surcharge of intensity q is located above the backfill, as shown in Figure 7.13, the active force, Pa, can be calculated as Pa 5 12KageqH 2 c Eq. (7.25) (7.27) where geq 5 g 1 c sinb 2q da b sin (b 1 a) H (7.28) Table 7.3 Values of Ka [Eq. (7.26)] for b 5 90° and a 5 0° d9 (deg) f9 (deg) 0 5 10 15 20 25 28 30 32 34 36 38 40 42 0.3610 0.3333 0.3073 0.2827 0.2596 0.2379 0.2174 0.1982 0.3448 0.3189 0.2945 0.2714 0.2497 0.2292 0.2098 0.1916 0.3330 0.3085 0.2853 0.2633 0.2426 0.2230 0.2045 0.1870 0.3251 0.3014 0.2791 0.2579 0.2379 0.2190 0.2011 0.1841 0.3203 0.2973 0.2755 0.2549 0.2354 0.2169 0.1994 0.1828 0.3186 0.2956 0.2745 0.2542 0.2350 0.2167 0.1995 0.1831 7.5 Coulomb’s Active Earth Pressure 343 Table 7.4 Values of Ka [from Eq. (7.26)] for dr 5 23 fr b (deg) a (deg) f9 (deg) 90 85 80 75 70 65 0 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 28 0.3213 0.3091 0.2973 0.2860 0.2750 0.2645 0.2543 0.2444 0.2349 0.2257 0.2168 0.2082 0.1998 0.1918 0.1840 0.3431 0.3295 0.3165 0.3039 0.2919 0.2803 0.2691 0.2583 0.2479 0.2379 0.2282 0.2188 0.2098 0.2011 0.1927 0.3702 0.3548 0.3400 0.3259 0.3123 0.2993 0.2868 0.2748 0.2633 0.2522 0.2415 0.2313 0.2214 0.2119 0.2027 0.4065 0.3588 0.3467 0.3349 0.3235 0.3125 0.3019 0.2916 0.2816 0.2719 0.2626 0.2535 0.2447 0.2361 0.2278 0.2197 0.3845 0.3709 0.3578 0.3451 0.3329 0.3211 0.3097 0.2987 0.2881 0.2778 0.2679 0.2582 0.2489 0.2398 0.2311 0.4164 0.4007 0.3857 0.3713 0.3575 0.3442 0.3314 0.3190 0.3072 0.2957 0.2846 0.2740 0.2636 0.2537 0.2441 0.4585 0.4007 0.3886 0.3769 0.3655 0.3545 0.3439 0.3335 0.3235 0.3137 0.3042 0.2950 0.2861 0.2774 0.2689 0.2606 0.4311 0.4175 0.4043 0.3916 0.3792 0.3673 0.3558 0.3446 0.3338 0.3233 0.3131 0.3033 0.2937 0.2844 0.2753 0.4686 0.4528 0.4376 0.4230 0.4089 0.3953 0.3822 0.3696 0.3574 0.3456 0.3342 0.3231 0.3125 0.3021 0.2921 0.5179 0.4481 0.4362 0.4245 0.4133 0.4023 0.3917 0.3813 0.3713 0.3615 0.3520 0.3427 0.3337 0.3249 0.3164 0.3080 0.4843 0.4707 0.4575 0.4447 0.4324 0.4204 0.4088 0.3975 0.3866 0.3759 0.3656 0.3556 0.3458 0.3363 0.3271 0.5287 0.5128 0.4974 0.4826 0.4683 0.4545 0.4412 0.4283 0.4158 0.4037 0.3920 0.3807 0.3697 0.3590 0.3487 0.5868 0.5026 0.4908 0.4794 0.4682 0.4574 0.4469 0.4367 0.4267 0.4170 0.4075 0.3983 0.3894 0.3806 0.3721 0.3637 0.5461 0.5325 0.5194 0.5067 0.4943 0.4823 0.4707 0.4594 0.4484 0.4377 0.4273 0.4172 0.4074 0.3978 0.3884 0.5992 0.5831 0.5676 0.5526 0.5382 0.5242 0.5107 0.4976 0.4849 0.4726 0.4607 0.4491 0.4379 0.4270 0.4164 0.6685 0.5662 0.5547 0.5435 0.5326 0.5220 0.5117 0.5017 0.4919 0.4824 0.4732 0.4641 0.4553 0.4468 0.4384 0.4302 0.6190 0.6056 0.5926 0.5800 0.5677 0.5558 0.5443 0.5330 0.5221 0.5115 0.5012 0.4911 0.4813 0.4718 0.4625 0.6834 0.6672 0.6516 0.6365 0.6219 0.6078 0.5942 0.5810 0.5682 0.5558 0.5437 0.5321 0.5207 0.5097 0.4990 0.7670 5 10 15 (continued) 344 Chapter 7: Lateral Earth Pressure Table 7.4 (Continued) b (deg) a (deg) f9 (deg) 90 85 80 75 70 65 20 29 30 31 32 33 34 35 36 37 38 39 40 41 42 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 0.3881 0.3707 0.3541 0.3384 0.3234 0.3091 0.2954 0.2823 0.2698 0.2578 0.2463 0.2353 0.2247 0.2146 0.4602 0.4364 0.4142 0.3935 0.3742 0.3559 0.3388 0.3225 0.3071 0.2925 0.2787 0.2654 0.2529 0.2408 0.2294 0.4397 0.4219 0.4049 0.3887 0.3732 0.3583 0.3442 0.3306 0.3175 0.3050 0.2929 0.2813 0.2702 0.2594 0.5205 0.4958 0.4728 0.4513 0.4311 0.4121 0.3941 0.3771 0.3609 0.3455 0.3308 0.3168 0.3034 0.2906 0.2784 0.4987 0.4804 0.4629 0.4462 0.4303 0.4150 0.4003 0.3862 0.3726 0.3595 0.3470 0.3348 0.3231 0.3118 0.5900 0.5642 0.5403 0.5179 0.4968 0.4769 0.4581 0.4402 0.4233 0.4071 0.3916 0.3768 0.3626 0.3490 0.3360 0.5672 0.5484 0.5305 0.5133 0.4969 0.4811 0.4659 0.4513 0.4373 0.4237 0.4106 0.3980 0.3858 0.3740 0.6714 0.6445 0.6195 0.5961 0.5741 0.5532 0.5335 0.5148 0.4969 0.4799 0.4636 0.4480 0.4331 0.4187 0.4049 0.6483 0.6291 0.6106 0.5930 0.5761 0.5598 0.5442 0.5291 0.5146 0.5006 0.4871 0.4740 0.4613 0.4491 0.7689 0.7406 0.7144 0.6898 0.6666 0.6448 0.6241 0.6044 0.5856 0.5677 0.5506 0.5342 0.5185 0.5033 0.4888 0.7463 0.7265 0.7076 0.6895 0.6721 0.6554 0.6393 0.6238 0.6089 0.5945 0.5805 0.5671 0.5541 0.5415 0.8880 0.8581 0.8303 0.8043 0.7799 0.7569 0.7351 0.7144 0.6947 0.6759 0.6579 0.6407 0.6242 0.6083 0.5930 Table 7.5 Values of Ka [from Eq. (7.26)] for dr 5 fr>2 b (deg) a (deg) f9 (deg) 90 85 80 75 70 65 0 28 29 30 31 32 33 34 35 36 0.3264 0.3137 0.3014 0.2896 0.2782 0.2671 0.2564 0.2461 0.2362 0.3629 0.3502 0.3379 0.3260 0.3145 0.3033 0.2925 0.2820 0.2718 0.4034 0.3907 0.3784 0.3665 0.3549 0.3436 0.3327 0.3221 0.3118 0.4490 0.4363 0.4241 0.4121 0.4005 0.3892 0.3782 0.3675 0.3571 0.5011 0.4886 0.4764 0.4645 0.4529 0.4415 0.4305 0.4197 0.4092 0.5616 0.5492 0.5371 0.5253 0.5137 0.5025 0.4915 0.4807 0.4702 7.5 Coulomb’s Active Earth Pressure 345 Table 7.5 (Continued) b (deg) a (deg) 5 10 15 f9 (deg) 90 85 80 75 70 65 37 38 39 40 41 42 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 28 29 30 31 32 33 34 35 0.2265 0.2172 0.2081 0.1994 0.1909 0.1828 0.3477 0.3337 0.3202 0.3072 0.2946 0.2825 0.2709 0.2596 0.2488 0.2383 0.2282 0.2185 0.2090 0.1999 0.1911 0.3743 0.3584 0.3432 0.3286 0.3145 0.3011 0.2881 0.2757 0.2637 0.2522 0.2412 0.2305 0.2202 0.2103 0.2007 0.4095 0.3908 0.3730 0.3560 0.3398 0.3244 0.3097 0.2956 0.2620 0.2524 0.2431 0.2341 0.2253 0.2168 0.3879 0.3737 0.3601 0.3470 0.3342 0.3219 0.3101 0.2986 0.2874 0.2767 0.2662 0.2561 0.2463 0.2368 0.2276 0.4187 0.4026 0.3872 0.3723 0.3580 0.3442 0.3309 0.3181 0.3058 0.2938 0.2823 0.2712 0.2604 0.2500 0.2400 0.4594 0.4402 0.4220 0.4046 0.3880 0.3721 0.3568 0.3422 0.3017 0.2920 0.2825 0.2732 0.2642 0.2554 0.4327 0.4185 0.4048 0.3915 0.3787 0.3662 0.3541 0.3424 0.3310 0.3199 0.3092 0.2988 0.2887 0.2788 0.2693 0.4688 0.4525 0.4368 0.4217 0.4071 0.3930 0.3793 0.3662 0.3534 0.3411 0.3292 0.3176 0.3064 0.2956 0.2850 0.5159 0.4964 0.4777 0.4598 0.4427 0.4262 0.4105 0.3953 0.3469 0.3370 0.3273 0.3179 0.3087 0.2997 0.4837 0.4694 0.4556 0.4422 0.4292 0.4166 0.4043 0.3924 0.3808 0.3695 0.3585 0.3478 0.3374 0.3273 0.3174 0.5261 0.5096 0.4936 0.4782 0.4633 0.4489 0.4350 0.4215 0.4084 0.3957 0.3833 0.3714 0.3597 0.3484 0.3375 0.5812 0.5611 0.5419 0.5235 0.5059 0.4889 0.4726 0.4569 0.3990 0.3890 0.3792 0.3696 0.3602 0.3511 0.5425 0.5282 0.5144 0.5009 0.4878 0.4750 0.4626 0.4505 0.4387 0.4272 0.4160 0.4050 0.3944 0.3840 0.3738 0.5928 0.5761 0.5599 0.5442 0.5290 0.5143 0.5000 0.4862 0.4727 0.4597 0.4470 0.4346 0.4226 0.4109 0.3995 0.6579 0.6373 0.6175 0.5985 0.5803 0.5627 0.5458 0.5295 0.4599 0.4498 0.4400 0.4304 0.4209 0.4177 0.6115 0.5972 0.5833 0.5698 0.5566 0.5437 0.5312 0.5190 0.5070 0.4954 0.4840 0.4729 0.4620 0.4514 0.4410 0.6719 0.6549 0.6385 0.6225 0.6071 0.5920 0.5775 0.5633 0.5495 0.5361 0.5230 0.5103 0.4979 0.4858 0.4740 0.7498 0.7284 0.7080 0.6884 0.6695 0.6513 0.6338 0.6168 (continued) 346 Chapter 7: Lateral Earth Pressure Table 7.5 (Continued) b (deg) a (deg) 20 f9 (deg) 90 85 80 75 70 65 36 37 38 39 40 41 42 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 0.2821 0.2692 0.2569 0.2450 0.2336 0.2227 0.2122 0.4614 0.4374 0.4150 0.3941 0.3744 0.3559 0.3384 0.3218 0.3061 0.2911 0.2769 0.2633 0.2504 0.2381 0.2263 0.3282 0.3147 0.3017 0.2893 0.2773 0.2657 0.2546 0.5188 0.4940 0.4708 0.4491 0.4286 0.4093 0.3910 0.3736 0.3571 0.3413 0.3263 0.3120 0.2982 0.2851 0.2725 0.3807 0.3667 0.3531 0.3401 0.3275 0.3153 0.3035 0.5844 0.5586 0.5345 0.5119 0.4906 0.4704 0.4513 0.4331 0.4157 0.3991 0.3833 0.3681 0.3535 0.3395 0.3261 0.4417 0.4271 0.4130 0.3993 0.3861 0.3733 0.3609 0.6608 0.6339 0.6087 0.5851 0.5628 0.5417 0.5216 0.5025 0.4842 0.4668 0.4500 0.4340 0.4185 0.4037 0.3894 0.5138 0.4985 0.4838 0.4695 0.4557 0.4423 0.4293 0.7514 0.7232 0.6968 0.6720 0.6486 0.6264 0.6052 0.5851 0.5658 0.5474 0.5297 0.5127 0.4963 0.4805 0.4653 0.6004 0.5846 0.5692 0.5543 0.5399 0.5258 0.5122 0.8613 0.8313 0.8034 0.7772 0.7524 0.7289 0.7066 0.6853 0.6649 0.6453 0.6266 0.6085 0.5912 0.5744 0.5582 Surcharge = q C A c = 0 H Pa B KaH sin sin ( + ) (a) (b) Figure 7.13 Coulomb’s active pressure with a surcharge on the backfill 7.5 Coulomb’s Active Earth Pressure 347 Example 7.6 Consider the retaining wall shown in Figure 7.12a. Given: H 5 4.6 m; unit weight of soil 5 16.5 kN>m3; angle of friction of soil 5 308; wall friction-angle, dr 5 23fr, soil cohesion, c9 5 0; a 5 0, and b 5 908. Calculate the Coulomb’s active force per unit length of the wall. Solution From Eq. (7.25) Pa 5 12gH 2Ka From Table 7.4, for a 5 08, b 5 908, f9 5 308, and d9 5 23fr 5208, Ka 5 0.297. Hence, Pa 5 12 (16.5) (4.6) 2 (0.297) 5 51.85 kN , m ■ Example 7.7 Refer to Figure 7.13a. Given: H ⫽ 6.1 m, ⬘ ⫽ 30°, ␦⬘ ⫽ 20°, ␣ ⫽ 5°,  ⫽ 85°, q ⫽ 96 kN/m2, and ␥ ⫽ 18 kN/m3. Determine Coulomb’s active force and the location of the line of action of the resultant Pa. Solution For  ⫽ 85°, ␣ ⫽ 5°, ␦⬘ ⫽ 20°, ⬘ ⫽ 30°, and Ka ⫽ 0.3578 (Table 7.4). From Eqs. (7.27) and (7.28), 2q sinb 1 1 1 pa 5 KageqH 2 5 Ka cg 1 d H 2 5 KagH 2 2 2 H sin (b 1 a) 2 (')'* sinb 1 KaHq c d Pa(1) sin (b 1 a) ('''')''''* Pa(2) 5 (0.5) (0.3578) (18) (6.1) 2 1 (0.3578) (6.1) (96) c sin85 d sin (85 1 5) 5 119.8 1 208.7 5 328.5 kN>m Location of the line of action of the resultant: Paz 5 Pa(1) H H 1 Pa(2) 3 2 or (119.8) a 6.1 6.1 b 1 (208.7) a b 3 2 z5 328.5 5 2.68 m (measured vertically from the bottom of the wall) ■ 348 Chapter 7: Lateral Earth Pressure 7.6 Lateral Earth Pressure Due to Surcharge In several instances, the theory of elasticity is used to determine the lateral earth pressure on unyielding retaining structures caused by various types of surcharge loading, such as line loading (Figure 7.14a) and strip loading (Figure 7.14b). According to the theory of elasticity, the stress at any depth, z, on a retaining structure caused by a line load of intensity q/unit length (Figure 7.14a) may be given as s5 2q a 2b pH (a2 1 b2 ) 2 (7.29) where ⫽ horizontal stress at depth z ⫽ bH (See Figure 7.14a for explanations of the terms a and b.) Line load q / unit length aH z = bH H (a) b z H a q / unit length P z (b) Figure 7.14 Lateral earth pressure caused by (a) line load and (b) strip load 7.6 Lateral Earth Pressure Due to Surcharge 349 However, because soil is not a perfectly elastic medium, some deviations from Eq. (7.29) may be expected. The modified forms of this equation generally accepted for use with soils are as follows: s5 4a a2b pH (a2 1 b2 ) for a . 0.4 (7.30) s5 q 0.203b H (0.16 1 b2 ) 2 for a # 0.4 (7.31) and Figure 7.14b shows a strip load with an intensity of q/unit area located at a distance b⬘ from a wall of height H. Based on the theory of elasticity, the horizontal stress, , at any depth z on a retaining structure is s5 q (b 2 sin b cos 2a) p (7.32) (The angles ␣ and  are defined in Figure 7.14b.) However, in the case of soils, the right-hand side of Eq. (7.32) is doubled to account for the yielding soil continuum, or s5 2q (b 2 sinb cos 2a) p (7.33) The total force per unit length (P) due to the strip loading only (Jarquio, 1981) may be expressed as P5 q 3H(u2 2 u1 )4 90 (7.34) where u1 5 tan2 a b9 b H u2 5 tan21 a (deg) a 9 1 b9 b (deg) H (7.35) (7.36) The location z– (see Figure 7.14b) of the resultant force, P, can be given as z5H2 c H 2 (u2 2 u1 ) 1 (R 2 Q) 2 57.3arH d 2H(u2 2 u1 ) (7.37) 350 Chapter 7: Lateral Earth Pressure where R ⫽ (a⬘ ⫹ b⬘)2(90 ⫺ 2) (7.38) Q ⫽ b⬘ (90 ⫺ 1) (7.39) 2 Example 7.8 Refer to Figure 7.14b. Here, a⬘ ⫽ 2 m, b⬘ ⫽ 1 m, q ⫽ 40 kN/m2, and H ⫽ 6 m. Determine the total force on the wall (kN/m) caused by the strip loading only. Solution From Eqs. (7.35) and (7.38), 1 u1 5 tan21 a b 5 9.46° 6 211 u2 5 tan21 a b 5 26.57° 6 From Eq. (7.34) P5 q 40 3H(u2 2 u1 )4 5 36(26.57 2 9.46)4 5 45.63 kN>m 90 90 ■ Example 7.9 Refer to Example 7.8. Determine the location of the resultant –z . Solution From Eqs. (7.38) and (7.39), R ⫽ (a⬘ ⫹ b⬘)2(90 ⫺ 2) ⫽ (2 ⫹ 1)2(90 ⫺ 26.57) ⫽ 570.87 Q ⫽ b⬘2(90 ⫺ 1) ⫽ (1)2(90 ⫺ 9.46) ⫽ 80.54 From Eq. (7.37), z5H2 c 562 c 7.7 H 2 (u2 2 u1 ) 1 (R 2 Q) 2 57.3arH d 2H(u2 2 u1 ) (6) 2 (26.57 2 9.46) 1 (570.87 2 80.54) 2 (57.3) (2) (6) d 5 3.96 m (2) (6) (26.57 2 9.46) ■ Active Earth Pressure for Earthquake Conditions Coulomb’s active earth pressure theory (see Section 7.5) can be extended to take into account the forces caused by an earthquake. Figure 7.15 shows a condition of active pressure with a granular backfill (c9 5 0). Note that the forces acting on the soil failure wedge in Figure 7.15 7.7 Active Earth Pressure for Earthquake Conditions 351 kW Granular backfill c = 0 khW W H Pae Figure 7.15 Derivation of Eq. (7.42) are essentially the same as those shown in Figure 7.12a with the addition of khW and kvW in the horizontal and vertical direction respectively; kh and kv may be defined as kh 5 horizontal earthquake acceleration component acceleration due to gravity, g (7.40) kv 5 vertical earthquake acceleration component acceleration due to gravity, g (7.41) As in Section 7.5, the relation for the active force per unit length of the wall (Pae) can be determined as Pae 5 12gH 2 (1 2 kv )Kae (7.42) where Kae 5 active earth pressure coefficient 5 sin2 (fr 1 b 2 ur) cos ur sin2 b sin(b 2 ur 2 dr) B1 1 sin (fr 1 dr)sin (fr 2 ur 2 a) 2 R Å sin (b 2 dr 2 ur)sin (a 1 b) (7.43) ur 5 tan21 c kh d (1 2 kv ) (7.44) 352 Chapter 7: Lateral Earth Pressure Note that for no earthquake condition kh 5 0, kv 5 0, and ur 5 0 Hence Kae 5 Ka [as given by Eq. (7.26)]. Some values of Kae for b 5 908 and kv 5 0 are given in Table 7.6. The magnitude of Pae as given in Eq. (7.42) also can be determined as (Seed and Whitman, 1970), sin2b9 1 Pae 5 gH 2 (1 2 kv ) 3Ka (b9,a9 )4 a b 2 cosu9sin2b (7.45) where ⬘ ⫽  ⫺ ⬘ (7.46) ␣⬘ ⫽ ⬘ ⫹ ␣ (7.47) Ka(⬘,␣⬘) ⫽ Coulomb’s active earth-pressure coefficient on a wall with a back face inclination of ⬘ with the horizontal and with a back fill inclined at an angle ␣⬘ with the horizontal (such as Tables 7.4 and 7.5) Equation (7.42) is usually referred to as the Mononobe–Okabe solution. Unlike the case shown in Figure 7.12a, the resultant earth pressure in this situation, as calculated by Eq. (7.42) does not act at a distance of H>3 from the bottom of the wall. The following procedure may be used to obtain the location of the resultant force Pae: Step 1. Calculate Pae by using Eq. (7.42) Step 2. Calculate Pa by using Eq. (7.25) Step 3. Calculate DPae 5 Pae 2 Pa (7.48) Step 4. Assume that Pa acts at a distance of H>3 from the bottom of the wall (Figure 7.16) Pae Pa H 0.6 H H/3 Figure 7.16 Determining the line of action of Pae 7.7 Active Earth Pressure for Earthquake Conditions 353 Table 7.6 Values of Kae [Eq. (7.43)] for b 5 908 and kv 5 0 f9(deg) 353 kh d9 (deg) a(deg) 28 30 35 40 45 0.1 0.2 0.3 0.4 0.5 0 0 0.427 0.508 0.611 0.753 1.005 0.397 0.473 0.569 0.697 0.890 0.328 0.396 0.478 0.581 0.716 0.268 0.382 0.400 0.488 0.596 0.217 0.270 0.334 0.409 0.500 0.1 0.2 0.3 0.4 0.5 0 5 0.457 0.554 0.690 0.942 — 0.423 0.514 0.635 0.825 — 0.347 0.424 0.522 0.653 0.855 0.282 0.349 0.431 0.535 0.673 0.227 0.285 0.356 0.442 0.551 0.1 0.2 0.3 0.4 0.5 0 10 0.497 0.623 0.856 — — 0.457 0.570 0.748 — — 0.371 0.461 0.585 0.780 — 0.299 0.375 0.472 0.604 0.809 0.238 0.303 0.383 0.486 0.624 0.1 0.2 0.3 0.4 0.5 fr>2 0 0.396 0.485 0.604 0.778 1.115 0.368 0.452 0.563 0.718 0.972 0.306 0.380 0.474 0.599 0.774 0.253 0.319 0.402 0.508 0.648 0.207 0.267 0.340 0.433 0.522 0.1 0.2 0.3 0.4 0.5 fr>2 5 0.428 0.537 0.699 1.025 — 0.396 0.497 0.640 0.881 — 0.326 0.412 0.526 0.690 0.962 0.268 0.342 0.438 0.568 0.752 0.218 0.283 0.367 0.475 0.620 0.1 0.2 0.3 0.4 0.5 fr>2 10 0.472 0.616 0.908 — — 0.433 0.562 0.780 — — 0.352 0.454 0.602 0.857 — 0.285 0.371 0.487 0.656 0.944 0.230 0.303 0.400 0.531 0.722 0.1 0.2 0.3 0.4 0.5 2 3 fr 0 0.393 0.486 0.612 0.801 1.177 0.366 0.454 0.572 0.740 1.023 0.306 0.384 0.486 0.622 0.819 0.256 0.326 0.416 0.533 0.693 0.212 0.276 0.357 0.462 0.600 0.1 0.2 0.3 0.4 0.5 2 3 fr 5 0.427 0.541 0.714 1.073 — 0.395 0.501 0.655 0.921 — 0.327 0.418 0.541 0.722 1.034 0.271 0.350 0.455 0.600 0.812 0.224 0.294 0.386 0.509 0.679 0.1 0.2 0.3 0.4 0.5 2 3 fr 10 0.472 0.625 0.942 — — 0.434 0.570 0.807 — — 0.354 0.463 0.624 0.909 — 0.290 0.381 0.509 0.699 1.037 0.237 0.317 0.423 0.573 0.800 354 Chapter 7: Lateral Earth Pressure Step 5. Assume that DPae acts at a distance of 0.6H from the bottom of the wall (Figure 7.16) Step 6. Calculate the location of the resultant as z5 (0.6H) (DPae ) 1 a H b (Pa ) 3 Pae Example 7.10 Refer to Figure 7.17. For kv 5 0 and kh 5 0.3, determine: a. Pae using Eq. (7.45) b. The location of the resultant, z, from the bottom of the wall 35 16.51 kN/m3 17.5 3.05 m Figure 7.17 Solution Part a From Eq. (7.44), u9 5 tan21 a kh 0.3 b 5 tan21 a b 5 16.7° 1 2 kv 120 From Eqs. (7.46) and (7.47), ⬘ ⫽  ⫺ ⬘ ⫽ 90 ⫺ 16.7 ⫽ 73.3° ␣⬘ ⫽ ⬘ ⫹ ␣ ⫽ 16.7 ⫹ 0 ⫽ 16.7° 17.5 d9 5 5 0.5 9 35 f (7.49) 7.8 Active Pressure for Wall Rotation about the Top: Braced Cut 355 We will refer to Table 7.5. For ⬘ ⫽ 35°, ␦⬘/⬘ ⫽ 0.5, ⬘ ⫽ 73.3°, and ␣⬘ ⫽ 16.7°, the value of Ka(⬘,␣⬘) ⬇ 0.495. Thus, from Eq. (7.45), sin2br 1 gH 2 (1 2 kv ) 3Ka (br,ar)4 a b 2 cos ursin2b 1 sin273.3 5 (16.51) (3.05) 2 (1 2 0) (0.495) a b 5 36.4 kN>m 2 cos16.7 sin290 Pae 5 Part b From Eq. (7.25), Pa 5 12gH 2Ka From Eq. (7.26) with d9 5 17.58, b 5 908, and a 5 08, Ka < 0.246 (Table 7.5). Pa 5 12 (16.51) (3.05) 2 (0.246) 5 18.89 kN>m DPae 5 Pae 2 Pa 5 36.32 2 18.89 5 17.43 kN>m From Eq. (7.49), z5 5 7.8 (0.6H) ( DPae ) 1 (H>3) (Pa ) Pae 3(0.6) (3.05) 4 (17.43) 1 (3.05>3) (18.89) 36.32 5 1.41 m ■ Active Pressure for Wall Rotation about the Top: Braced Cut In the preceding sections, we have seen that a retaining wall rotates about its bottom. (See Figure 7.18a.) With sufficient yielding of the wall, the lateral earth pressure is approximately equal to that obtained by Rankine’s theory or Coulomb’s theory. In contrast to retaining walls, braced cuts show a different type of wall yielding. (See Figure 7.18b.) In this case, deformation of the wall gradually increases with the depth of excavation. The variation of the amount of deformation depends on several factors, such as the type of soil, the depth of excavation, and the workmanship involved. However, with very little wall yielding at the top of the cut, the lateral earth pressure will be close to the at-rest pressure. At the bottom of the wall, with a much larger degree of yielding, the lateral earth pressure will be substantially lower than the Rankine active earth pressure. As a result, the distribution of lateral earth pressure will vary substantially in comparison to the linear distribution assumed in the case of retaining walls. The total lateral force per unit length of the wall, Pa , imposed on a wall may be evaluated theoretically by using Terzaghi’s (1943) general wedge theory. (See Figure 7.19.) The failure surface is assumed to be the arc of a logarithmic spiral, defined as r 5 roeu tan fr where fr 5 effective angle of friction of soil. (7.50) 356 Chapter 7: Lateral Earth Pressure Strut (a) (b) Figure 7.18 Nature of yielding of walls: (a) retaining wall; (b) braced cut Center of log spiral c 90 Lateral pressure, a b c Ca naH Pa W H C a R Figure 7.19 Braced cut analysis by general wedge theory: wall rotation about top Arc of log spiral In the figure, H is the height of the cut, and the unit weight, angle of friction, and cohesion of the soil are equal to g, fr, and cr, respectively. Following are the forces per unit length of the cut acting on the trial failure wedge: 1. 2. 3. 4. 5. Weight of the wedge, W Resultant of the normal and shear forces along ab, R Cohesive force along ab, C Adhesive force along ac, Ca Pa , which is the force acting a distance naH from the bottom of the wall and is inclined at an angle d9 to the horizontal The adhesive force is Ca 5 carH (7.51) where car 5 unit adhesion. A detailed outline for the evaluation of Pa is beyond the scope of this text; those interested should check a soil mechanics text for more information. Kim and Preber (1969) provided tabulated values of Pa> 12gH 2 determined by using the principles of general wedge theory. Table 7.7 gives the variation of Pa>0.5gH 2 for granular soil backfill obtained using the general wedge theory. 357 7.9 Active Earth Pressure for Translation of Retaining Wall—Granular Backfil Table 7.7 Active Pressure for Wall Rotation—General Wedge Theory (Granular Soil Backfill) Soil friction angle, f9(deg) 25 Pa , 0.5 gH 2 d9 , f9 na 5 0.3 na 5 0.4 na 5 0.5 na 5 0.6 0 0.371 0.345 0.342 0.344 0.405 0.376 0.373 0.375 0.447 0.413 0.410 0.413 0.499 0.460 0.457 0.461 0.304 0.282 0.281 0.289 0.330 0.306 0.305 0.313 0.361 0.334 0.332 0.341 0.400 0.386 0.367 0.377 0.247 0.231 0.232 0.243 0.267 0.249 0.249 0.262 0.290 0.269 0.270 0.289 0.318 0.295 0.296 0.312 0.198 0.187 0.190 0.197 0.213 0.200 0.204 0.211 0.230 0.216 0.220 0.228 0.252 0.235 0.239 0.248 0.205 0.149 0.153 0.173 0.220 0.159 0.164 0.184 0.237 0.171 0.176 0.198 0.259 0.185 0.196 0.215 1⁄2 2⁄3 1 30 0 1⁄2 2⁄3 1 35 0 1⁄2 2⁄3 1 40 0 1⁄2 2⁄3 1 45 0 1⁄2 2⁄3 1 7.9 Active Earth Pressure for Translation of Retaining Wall—Granular Backfill Under certain circumstances, retaining walls may undergo lateral translation, as shown in Figure 7.20. A solution to the distribution of active pressure for this case was provided by Dubrova (1963) and was also described by Harr (1966). The solution of Dubrova assumes the validity of Coulomb’s solution [Eqs. (7.25) and (7.26)]. In order to understand this procedure, let us consider a vertical wall with a horizontal granular backfill (Figure 7.21). For rotation about the top of the wall, the resultant R of the normal and shear forces along the rupture line AC is inclined at an angle f9 to the normal drawn to AC. According to Dubrova there exists infinite number of quasi-rupture lines such as A9C9, A0C0, . . . for which the resultant force R is inclined at an angle c, where c5 frz H (7.52) Now, refer to Eqs. (7.25) and (7.26) for Coulomb’s active pressure. For b 5 908 and a 5 0, the relationship for Coulomb’s active force can also be rewritten as 2 Pa 5 g D 2 cos dr H T 1 1 (tan2 fr 1 tan fr tan dr) 0.5 cos fr (7.53) 358 Chapter 7: Lateral Earth Pressure B C z A R H C C R Granular backfill c 0 A R A Figure 7.20 Lateral translation of retaining wall Figure 7.21 Quasi-rupture lines behind a retaining wall The force against the wall at any z is then given as 2 Pa 5 g z D T 2 cos dr 1 2 0.5 1 (tan c 1 tan c tan dr) cos c (7.54) The active pressure at any depth z for wall rotation about the top is sar (z) 5 dPa g z cos2 c z2fr cos2 c < c 2 (sin c 1 m) d (7.55) 2 dz cos dr (1 1 m sin c) H(1 1 m sin c) where m 5 a1 1 tan dr 0.5 b . tan c (7.56) For frictionless walls, d9 5 0 and Eq. (7.55) simplifies to sar (z) 5 g tan2 a45 2 c frz2 b az 2 b 2 H cos c (7.57) For wall rotation about the bottom, a similar expression can be found in the form sar (z) 5 2 gz cos fr a b cos dr 1 1 m sin fr (7.58) For translation of the wall, the active pressure can then be taken as sar (z) translation 5 12 3sar (z) rotation about top 1 sar (z) rotation about bottom4 (7.59) 7.9 Active Earth Pressure for Translation of Retaining Wall—Granular Backfil 359 Example 7.11 Consider a frictionless wall 5 m high. For the granular backfill, g 5 17.3 kN/m3 and f9 5 368. Calculate and plot the variation of sa(z) for a translation mode of the wall movement. Solution For a frictionless wall, d9 5 0. Hence, m is equal to one [Eq. (7.56)]. So for rotation about the top, from Eq. (7.57), sar (z) 5 sa(1) r 5 g tan2 a45 2 frz b Dz 2 2H frz2 T frz b H cos a H For rotation about the bottom, from Eq. (7.58), sar (z) 5 sa(2) r 5 gz a sar (z) translation 5 2 cos fr b 1 1 sin fr sa(1) r 1 sa(2) r 2 The following table can now be prepared with g 5 17.3 kN/m3, f9 5 368, and H 5 5 m. z (m) sa9(1) (kN , m2) sa9(2) (kN , m2) s9a(z) translation (kN , m2) 0 1.25 2.5 3.75 5.0 0 13.26 15.26 11.48 5.02 0 5.62 11.24 16.86 22.48 0 9.44 13.25 14.17 13.75 The plot of sa(z) versus z is shown in Figure 7.22 a(z) translation (kN/m2) 0 5 10 15 0 z (m) 1.25 2.5 3.75 5 Figure 7.22 ■ 360 Chapter 7: Lateral Earth Pressure Passive Pressure 7.10 Rankine Passive Earth Pressure Figure 7.23a shows a vertical frictionless retaining wall with a horizontal backfill. At depth z, the effective vertical pressure on a soil element is sor 5 gz. Initially, if the wall does not yield at all, the lateral stress at that depth will be shr 5 Kosor. This state of stress is illustrated by the Mohr’s circle a in Figure 7.23b. Now, if the wall is pushed into the soil mass by an amount Dx, as shown in Figure 7.23a, the vertical stress at depth z will stay the same; however, the horizontal stress will increase. Thus, shr will be greater than Kosor . The state of stress can now be represented by the Mohr’s circle b in Figure 7.23b. If the wall moves farther inward (i.e., Dx is increased still more), the stresses at depth z will ultimately reach the state represented by Mohr’s circle c. Note that this Mohr’s circle touches the Mohr–Coulomb failure envelope, which implies that the soil behind the wall will fail by being pushed upward. The horizontal stress, shr , at this point is referred to as the Rankine passive pressure, or shr 5 spr . For Mohr’s circle c in Figure 7.23b, the major principal stress is spr , and the minor principal stress is sor . Substituting these quantities into Eq. (1.87) yields spr 5 sor tan2 ¢ 45 1 fr fr ≤ 1 2crtan ¢45 1 ≤ 2 2 (7.60) Now, let Kp 5 Rankine passive earth pressure coefficient 5 tan2 ¢45 1 fr ≤ 2 (7.61) Then, from Eq. (7.60), we have spr 5 sor Kp 1 2cr"Kp (7.62) Equation (7.62) produces (Figure 7.23c), the passive pressure diagram for the wall shown in Figure 7.23a. Note that at z 5 0, sor 5 0 and spr 5 2cr"Kp and at z 5 H, sor 5 gH and spr 5 gHKp 1 2cr"Kp The passive force per unit length of the wall can be determined from the area of the pressure diagram, or Pp 5 12gH 2Kp 1 2crH"Kp (7.63) The approximate magnitudes of the wall movements, Dx, required to develop failure under passive conditions are as follows: 7.10 Rankine Passive Earth Pressure Wall movement for passive condition, D x Soil type Dense sand Loose sand Stiff clay Soft clay 0.005H 0.01H 0.01H 0.05H Direction of wall movement x o z 45 /2 45 /2 z c H h Rotation about this point (a) Shear stress s c c n ta c a Effective normal stress h p b h Koo h o (b) 2c Kp H Kp H 2c Kp (c) Figure 7.23 Rankine passive pressure 361 362 Chapter 7: Lateral Earth Pressure If the backfill behind the wall is a granular soil (i.e., cr 5 0), then, from Eq. (7.63), the passive force per unit length of the wall will be Pp 5 1 gH 2Kp 2 (7.64) Example 7.12 A 3-m high wall is shown in Figure 7.24a. Determine the Rankine passive force per unit length of the wall. Solution For the top layer Kp(1) 5 tan2 a45 1 z 2m f1r b 5 tan2 (45 1 15) 5 3 2 15.72 kN/m3 1 30 c1 0 Groundwater table 1 112.49 94.32 1m sat 18.86 kN/m3 2 26 c2 10 kN/m2 2 3 0 (a) 135.65 kN/m2 (b) 4 Figure 7.24 From the bottom soil layer Kp(2) 5 tan2 a45 1 f2r b 5 tan2 (45 1 13) 5 2.56 2 spr 5 sor Kp 1 2cr"Kp where sor 5 effective vertical stress at z 5 0, sor 5 0, c1r 5 0, spr 5 0 at z 5 2 m, sor 5 (15.72) (2) 5 31.44 kN>m2, c1r 5 0 So, for the top soil layer spr 5 31.44Kp(1) 1 2(0)"Kp(1) 5 31.44(3) 5 94.32 kN>m2 9.81 kN/m2 7.11 Rankine Passive Earth Pressure: Vertical Backface and Inclined Backfill 363 At this depth, that is z 5 2 m, for the bottom soil layer spr 5 sor Kp(2) 1 2c2r "Kp(2) 5 31.44(2.56) 1 2(10)"2.56 5 80.49 1 32 5 112.49 kN>m2 Again, at z 5 3 m, sor 5 (15.72) (2) 1 (gsat 2 gw ) (1) 5 31.44 1 (18.86 2 9.81) (1) 5 40.49 kN>m2 Hence, spr 5 sor Kp(2) 1 2c2r "Kp(2) 5 40.49(2.56) 1 (2) (10) (1.6) 5 135.65 kN , m2 Note that, because a water table is present, the hydrostatic stress, u, also has to be taken into consideration. For z 5 0 to 2 m, u 5 0; z 5 3 m, u 5 (1)(gw) 5 9.81 kN>m2. The passive pressure diagram is plotted in Figure 6.24b. The passive force per unit length of the wall can be determined from the area of the pressure diagram as follows: Area no. 1 2 3 4 Area (12 ) (2)(94.32) (112.49)(1) (12 ) (1)(135.65 2 112.49) (12 ) (9.81)(1) 5 94.32 5 112.49 5 11.58 5 4.905 PP < 223.3 kN , m 7.11 ■ Rankine Passive Earth Pressure: Vertical Backface and Inclined Backfill Granular Soil For a frictionless vertical retaining wall (Figure 7.10) with a granular backfill (cr 5 0), the Rankine passive pressure at any depth can be determined in a manner similar to that done in the case of active pressure in Section 7.4. The pressure is spr 5 gzKp (7.65) Pp 5 12gH 2Kp (7.66) and the passive force is where Kp 5 cos a cos a 1 "cos2 a 2 cos2 fr cos a2"cos2 a 2 cos2 fr (7.67) 364 Chapter 7: Lateral Earth Pressure Table 7.8 Passive Earth Pressure Coefficient Kp [from Eq. (7.67)] S f9 (deg)S Ta (deg) 28 30 32 34 36 38 40 0 5 10 15 20 25 2.770 2.715 2.551 2.284 1.918 1.434 3.000 2.943 2.775 2.502 2.132 1.664 3.255 3.196 3.022 2.740 2.362 1.894 3.537 3.476 3.295 3.003 2.612 2.135 3.852 3.788 3.598 3.293 2.886 2.394 4.204 4.136 3.937 3.615 3.189 2.676 4.599 4.527 4.316 3.977 3.526 2.987 As in the case of the active force, the resultant force, Pp , is inclined at an angle a with the horizontal and intersects the wall at a distance H>3 from the bottom of the wall. The values of Kp (the passive earth pressure coefficient) for various values of a and fr are given in Table 7.8. c 9–f9 Soil If the backfill of the frictionless vertical retaining wall is a c9–f9 soil (see Figure 7.10), then (Mazindrani and Ganjali, 1997) sar 5 gzKp 5 gzKpr cos a (7.68) where 1 Kpr 5 d cos2 fr cr ≤cos fr sin fr gz t 21 cr 2 2 cr 2 2 2 2 1 4 cos a(cos a 2 cos fr) 1 4 ¢ ≤ cos fr 1 8¢ ≤cos a sin fr cos fr gz gz Å (7.69) 2 cos2 a 1 2 ¢ The variation of Kpr with fr, a, and cr>gz is given in Table 7.9 (Mazindrani and Ganjali, 1997). Table 7.9 Values of Kpr c9 , gz f9 (deg) a (deg) 0.025 0.050 0.100 0.500 15 0 5 10 15 0 5 10 15 0 5 10 15 1.764 1.716 1.564 1.251 2.111 2.067 1.932 1.696 2.542 2.499 2.368 2.147 1.829 1.783 1.641 1.370 2.182 2.140 2.010 1.786 2.621 2.578 2.450 2.236 1.959 1.917 1.788 1.561 2.325 2.285 2.162 1.956 2.778 2.737 2.614 2.409 3.002 2.971 2.880 2.732 3.468 3.435 3.339 3.183 4.034 3.999 3.895 3.726 20 25 7.12 Coulomb’s Passive Earth Pressure 365 Table 7.9 (Continued) c9 , gz 7.12 f9 (deg) a (deg) 0.025 0.050 0.100 0.500 30 0 5 10 15 3.087 3.042 2.907 2.684 3.173 3.129 2.996 2.777 3.346 3.303 3.174 2.961 4.732 4.674 4.579 4.394 Coulomb’s Passive Earth Pressure Coulomb (1776) also presented an analysis for determining the passive earth pressure (i.e., when the wall moves into the soil mass) for walls possessing friction (dr 5 angle of wall friction) and retaining a granular backfill material similar to that discussed in Section 7.5. To understand the determination of Coulomb’s passive force, Pp , consider the wall shown in Figure 7.25a. As in the case of active pressure, Coulomb assumed that the potential failure surface in soil is a plane. For a trial failure wedge of soil, such as ABC1 , the forces per unit length of the wall acting on the wedge are 1. The weight of the wedge, W 2. The resultant, R, of the normal and shear forces on the plane BC1 , and 3. The passive force, Pp Passive force Pp(min) Wall movement toward the soil C2 C3 C1 A Pp R H W Pp H/3 1 c 0 R W 1 B (a) Figure 7.25 Coulomb’s passive pressure (b) 366 Chapter 7: Lateral Earth Pressure Table 7.10 Values of Kp [from Eq. (7.71)] for b 5 90° and a 5 0° d9 (deg) f9 (deg) 0 5 10 15 20 15 20 25 30 35 40 1.698 2.040 2.464 3.000 3.690 4.600 1.900 2.313 2.830 3.506 4.390 5.590 2.130 2.636 3.286 4.143 5.310 6.946 2.405 3.030 3.855 4.977 6.854 8.870 2.735 3.525 4.597 6.105 8.324 11.772 Figure 7.25b shows the force triangle at equilibrium for the trial wedge ABC1 . From this force triangle, the value of Pp can be determined, because the direction of all three forces and the magnitude of one force are known. Similar force triangles for several trial wedges, such as ABC1, ABC2, ABC3, c, can be constructed, and the corresponding values of Pp can be determined. The top part of Figure 7.25a shows the nature of variation of the Pp values for different wedges. The minimum value of Pp in this diagram is Coulomb’s passive force, mathematically expressed as Pp 5 12 gH 2Kp (7.70) where Kp 5 Coulomb’s passive pressure coefficient 5 sin2 (b2fr) sin (fr 1 dr)sin (fr 1 a) 2 sin b sin (b 1 dr) B1 2 R Å sin (b 1 dr)sin (b 1 a) (7.71) 2 The values of the passive pressure coefficient, Kp , for various values of fr and dr are given in Table 7.10 (b 5 90°, a 5 0°). Note that the resultant passive force, Pp , will act at a distance H>3 from the bottom of the wall and will be inclined at an angle d9 to the normal drawn to the back face of the wall. 7.13 Comments on the Failure Surface Assumption for Coulomb’s Pressure Calculations Coulomb’s pressure calculation methods for active and passive pressure have been discussed in Sections 7.5 and 7.12. The fundamental assumption in these analyses is the acceptance of plane failure surface. However, for walls with friction, this assumption does not hold in practice. The nature of actual failure surface in the soil mass for active and passive pressure is shown in Figure 7.26a and b, respectively (for a vertical wall with a horizontal backfill). Note that the failure surface BC is curved and that the failure surface CD is a plane. Although the actual failure surface in soil for the case of active pressure is somewhat different from that assumed in the calculation of the Coulomb pressure, the results are not greatly different. However, in the case of passive pressure, as the value of dr increases, Coulomb’s 367 7.13 Comments on the Failure Surface Assumption for Coulomb’s Pressure Calculations A D 45 /2 2 H 3 45 /2 H Pa C B (a) C A D 45 /2 45 /2 2 H 3H Pp C B (b) Figure 7.26 Nature of failure surface in soil with wall friction: (a) active pressure; (b) passive pressure method of calculation gives increasingly erroneous values of Pp . This factor of error could lead to an unsafe condition because the values of Pp would become higher than the soil resistance. Several studies have been conducted to determine the passive force Pp , assuming that the curved portion BC in Figure 7.26b is an arc of a circle, an ellipse, or a logarithmic spiral. Shields and Tolunay (1973) analyzed the problem of passive pressure for a vertical wall with a horizontal granular soil backfill (cr 5 0). This analysis was done by considering the stability of the wedge ABCCr (see Figure 7.26b), using the method of slices and assuming BC as an arc of a logarthmic spiral. From Figure 7.26b, the passive force per unit length of the wall can be expressed as 1 Pp 5 KpgH 2 2 (7.72) The values of the passive earth-pressure coefficient, Kp , obtained by Shields and Tolunay are given in Figure 7.27. These are as good as any for design purposes. Solution by the Method of Triangular Slices Zhu and Qian (2000) used the method of triangular slices (such as in the zone of ABC in Fig. 7.28) to obtain the variation of Kp. According to this analysis Kp ⫽ Kp(␦⬘ ⫽ 0)R (7.73) where Kp ⫽ passive earth-pressure coefficient for a given value of , ␦⬘, and ⬘ Kp(␦⬘ ⫽ 0) ⫽ Kp for a given value of , ⬘ with ␦⬘ ⫽ 0 R ⫽ modification factor which is a function of ⬘, , ␦⬘/⬘ 368 Chapter 7: Lateral Earth Pressure 18 16 1 14 0.8 12 0.6 10 Kp 0.4 8 0.2 6 0 4 2 0 20 25 30 35 Soil friction angle, (deg) 40 45 Figure 7.27 Kp based on Shields and Tolunay’s analysis A C′ 45 – ′/2 45 – ′/2 Granular soil ′ H ′ H 3 C B Figure 7.28 Passive pressure solution by the method of triangular slices (Note: BC is arc of a logarthimic spiral) 7.13 Comments on the Failure Surface Assumption for Coulomb’s Pressure Calculations 369 The variations of Kp(␦⬘ ⫽ 0) are given in Table 7.11. The interpolated values of R are given in Table 7.12 Table 7.11 Variation of Kp(␦⬘ ⫽ 0) [see Eq. (7.73) and Figure 7.28]* (deg) ⬘(deg) 30 25 20 15 10 5 0 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 1.70 1.74 1.77 1.81 1.84 1.88 1.91 1.95 1.99 2.03 2.07 2.11 2.15 2.20 2.24 2.29 2.33 2.38 2.43 2.48 2.53 2.59 2.64 2.70 2.76 2.82 1.69 1.73 1.77 1.81 1.85 1.89 1.93 1.98 2.02 2.07 2.11 2.16 2.21 2.26 2.32 2.37 2.43 2.49 2.55 2.61 2.67 2.74 2.80 2.88 2.94 3.02 1.72 1.76 1.80 1.85 1.90 1.95 1.99 2.05 2.10 2.15 2.21 2.27 2.33 2.39 2.45 2.52 2.59 2.66 2.73 2.81 2.89 2.97 3.05 3.14 3.23 3.32 1.77 1.81 1.87 1.92 1.97 2.03 2.09 2.15 2.21 2.27 2.34 2.41 2.48 2.56 2.64 2.72 2.80 2.89 2.98 3.07 3.17 3.27 3.38 3.49 3.61 3.73 1.83 1.89 1.95 2.01 2.07 2.14 2.21 2.28 2.35 2.43 2.51 2.60 2.68 2.77 2.87 2.97 3.07 3.18 3.29 3.41 3.53 3.66 3.80 3.94 4.09 4.25 1.92 1.99 2.06 2.13 2.21 2.28 2.36 2.45 2.54 2.63 2.73 2.83 2.93 3.04 3.16 3.28 3.41 3.55 3.69 3.84 4.00 4.16 4.34 4.52 4.72 4.92 2.04 2.12 2.20 2.28 2.37 2.46 2.56 2.66 2.77 2.88 3.00 3.12 3.25 3.39 3.53 3.68 3.84 4.01 4.19 4.38 4.59 4.80 5.03 5.27 5.53 5.80 * Based on Zhu and Qian, 2000 Table 7.12 Variation of R [Eq. (7.73)] R for ⬘ (deg) (deg) ␦⬘/⬘ 30 35 40 45 0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.65 1.95 2.2 1.2 1.4 1.6 1.9 2.15 1.28 1.6 1.95 2.4 2.85 1.25 1.6 1.9 2.35 2.8 1.35 1.8 2.4 3.15 3.95 1.32 1.8 2.35 3.05 3.8 1.45 2.2 3.2 4.45 6.1 1.4 2.1 3.0 4.3 5.7 5 370 Chapter 7: Lateral Earth Pressure Table 7.12 (Continued) R for ⬘ (deg) (deg) ␦⬘/⬘ 30 35 40 45 10 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 1.15 1.35 1.6 1.8 2.05 1.15 1.35 1.55 1.8 2.0 1.15 1.35 1.5 1.8 1.9 1.2 1.5 1.85 2.25 2.65 1.2 1.5 1.8 2.2 2.6 1.2 1.45 1.8 2.1 2.4 1.3 1.7 2.25 2.9 3.6 1.3 1.65 2.2 2.8 3.4 1.3 1.65 2.1 2.6 3.2 1.4 2.0 2.9 4.0 5.3 1.35 1.95 2.7 3.8 4.95 1.35 1.9 2.6 3.55 4.8 15 20 7.14 Passive Pressure under Earthquake Conditions The relationship for passive earth pressure on a retaining wall with a granular backfill and under earthquake conditions was evaluated by Subba Rao and Choudhury (2005) by the method of limit equilibrium using the pseudo-static approach. Figure 7.29 shows the nature of failure surface in soil considered in this analysis. The passive pressure, Ppe,, can be expressed as Ppe 5 312gH 2Kpg(e) 4 1 cos dr (7.74) where Kpg(e) 5 passive earth-pressure coefficient in the normal direction to the wall. A D 45 /2 45 /2 H 2H 3 Ppe C Granular soil c 0 B Figure 7.29 Nature of failure surface in soil considered in the analysis to determine Ppe Problems 10 15 k 0 40 12 371 k kh 40 8 k 0 k kh 9 40 4 30 (e) 6 Kp Kp (e) 40 6 30 30 30 3 20 2 20 10 20 10 1 10 20 10 0.5 0 0 0 0.1 0.2 0.3 0.4 0.5 0 kh (a) Figure 7.30 Variation of Kpg(e) : (a) 0.1 0.2 0.3 0.4 0.5 kh (b) dr dr 5 1, (b) 5 0.5 fr fr Kpg(e) is a function of kh and kv that are, respectively, coefficient of horizontal and vertical acceleration due to earthquake. The variations of Kpg(e) for dr>fr 5 0.5 and 1 are shown in Figures 7.30a and b. The passive pressure Ppe will be inclined at an angle d9 to the back face of the wall and will act at a distance of H>3 above the bottom of the wall. Problems Refer to Figure 7.3a. Given: H 5 3.5 m, q 5 20 kN/m2, g 5 18.2 kN/m3 c9 5 0, and f9 5 358. Determine the at-rest lateral earth force per meter length of the wall. Also, find the location of the resultant. Use Eq. (7.4) and OCR = 1.5. 7.2 Use Eq. (7.3), Figure P7.2, and the following values to determine the at-rest lateral earth force per unit length of the wall. Also find the location of the resultant. H 5 5 m, H1 5 2 m, H2 5 3 m, g 5 15.5 kN>m3, gsat 5 18.5 kN>m3, f9 5 34°, c9 5 0, q 5 20 kN>m2, and OCR 5 1. 7.3 Refer to Figure 7.6a. Given the height of the retaining wall, H is 6.4 m; the backfill is a saturated clay with f 5 08, c 5 30.2 kN/m2, gsat 5 17.76 kN/m3, a. Determine the Rankine active pressure distribution diagram behind the wall. b. Determine the depth of the tensile crack, zc. c. Estimate the Rankine active force per foot length of the wall before and after the occurrence of the tensile crack. 7.1 372 Chapter 7: Lateral Earth Pressure q H1 z 1 c1 1 Groundwater table H H2 2 c2 2 Figure P7.2 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 A vertical retaining wall (Figure 7.6a) is 6.3 m high with a horizontal backfill. For the backfill, assume that g 5 17.9 kN>m3, f9 5 268, and c9 5 15 kN>m2. Determine the Rankine active force per unit length of the wall after the occurrence of the tensile crack. Refer to Problem 7.2. For the retaining wall, determine the Rankine active force per unit length of the wall and the location of the line of action of the resultant. Refer to Figure 7.10. For the retaining wall, H 5 6 m, fr 5 34°, a 5 10°, g 5 17 kN>m3, and c9 5 0. a. Determine the intensity of the Rankine active force at z 5 2 m, 4 m, and 6 m. b. Determine the Rankine active force per meter length of the wall and also the location and direction of the resultant. Refer to Figure 7.10. Given: H 5 6.7 m, g 5 18.08 kN/m3, f9 5 258, c9 5 12 kN/m2, and a 5 108. Calculate the Rankine active force per unit length of the wall after the occurrence of the tensile crack. Refer to Figure 7.12a. Given: H 5 3.66 m, g 5 16.5 kN/m3, f9 5 308, c9 5 0, and b 5 858. Determine the Coulomb’s active force per foot length of the wall and the location and direction of the resultant for the following cases: a. a 5 108 and d9 5 208 b. a 5 208 and d9 5 158 Refer to Figure 7.13a. Given H ⫽ 3.5 m, ␣ ⫽ 0,  ⫽ 85°, ␥ ⫽ 18 kN/m3, c⬘ ⫽ 0, f9 5 348, ␦⬘/⬘ ⫽ 0.5, and q ⫽ 30 kN/m2. Determine the Coulomb’s active force per unit length of the wall. Refer to Figure 7.14b. Given H ⫽ 3.3 m, a⬘ ⫽ 1 m, b⬘ ⫽ 1.5 m, and q ⫽ 25 kN/m2. Determine the lateral force per unit length of the unyielding wall caused by the surcharge loading only. Refer to Figure 7.15. Here, H 5 6 m, g 5 17 kN>m3, f9 5 358, d9 5 17.58, c9 5 0, a 5 108, and b 5 908. Determine the Coulomb’s active force for earthquake conditions (Pae) per meter length of the wall and the location and direction of the resultant. Given kh 5 0.2 and kv 5 0. References 8m 373 = 17.5 kN/m3 = 35 c = 0 Pa Figure P7.12 7.12 A retaining wall is shown in Figure P7.12. If the wall rotates about its top, determine the magnitude of the active force per unit length of the wall for na 5 0.3, 0.4, and 0.5. Assume dr>fr 5 0.5. 7.13 A vertical frictionless retaining wall is 6-m high with a horizontal granular backfill. Given: g 5 16 kN>m3 and f9 5 308. For the translation mode of the wall, calculate the active pressure at depths z 5 1.5 m, 3 m, 4.5 m, and 6 m. 7.14 Refer to Problem 7.3. a. Draw the Rankine passive pressure distribution diagram behind the wall. b. Estimate the Rankine passive force per foot length of the wall and also the location of the resultant. 7.15 In Figure 7.28, which shows a vertical retaining wall with a horizontal backfill, let H 5 4 m, u 5 108, g 5 16.5 kN>m3, f9 5 358, and d9 5 108. Based on Zhu and Qian’s work, what would be the passive force per meter length of the wall? 7.16 Consider a 4-m high retaining wall with a vertical back and horizontal granular backfill, as shown in Figure 7.29. Given: g 5 18 kN>m3, f9 5 408, c9 5 0, d9 5 208, kv 5 0 and kh 5 0.2. Determine the passive force Ppe per unit length of the wall taking the earthquake effect into consideration. References CHU, S. C. (1991). “Rankine Analysis of Active and Passive Pressures on Dry Sand,” Soils and Foundations, Vol. 31, No. 4, pp. 115–120. COULOMB, C. A. (1776). Essai sur une Application des Règles de Maximis et Minimum à quelques Problemes de Statique Relatifs à l’Architecture, Mem. Acad. Roy. des Sciences, Paris, Vol. 3, p. 38. DUBROVA, G. A. (1963). “Interaction of Soil and Structures,” Izd. Rechnoy Transport, Moscow. HARR, M. E. (1966). Fundamentals of Theoretical Soil Mechanics, McGraw-Hill, New York. JAKY, J. (1944). “The Coefficient of Earth Pressure at Rest,” Journal for the Society of Hungarian Architects and Engineers, October, pp. 355–358. JARQUIO, R. (1981). “Total Lateral Surcharge Pressure Due to Strip Load,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 107, No. GT10, pp. 1424–1428. KIM, J. S., and Preber, T. (1969). “Earth Pressure against Braced Excavations,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 96, No. 6, pp. 1581–1584. MAYNE, P. W., and Kulhawy, F. H. (1982). “Ko– OCR Relationships in Soil,” Journal of the Geotechnical Engineering Division, ASCE, Vol. 108, No. GT6, pp. 851– 872. 374 Chapter 7: Lateral Earth Pressure MAZINDRANI, Z. H., and Ganjali, M. H. (1997). “Lateral Earth Pressure Problem of Cohesive Backfill with Inclined Surface,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 123, No. 2, pp. 110 –112. SEED, H. B., and WHITMAN, R. V. (1970). “Design of Earth Retaining Structures for Dynamic Loads,” Proceedings, Specialty Conference on Lateral Stresses in the Ground and Design of Earth Retaining Structures, American Society of Civil Engineers, pp. 103–147. SHIELDS, D. H., and Tolunay, A. Z. (1973). “Passive Pressure Coefficients by Method of Slices,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 99, No. SM12, pp. 1043–1053. SUBBA RAO, K. S., and CHOUDHURY, D. (2005). “Seismic Passive Earth Pressures in Soil,” Journal of Geotechnical and Geoenvironmental Engineering, American Society of Civil Engineers, Vo. 131, No. 1, pp. 131–135. TERZAGHI, K. (1943). Theoretical Soil Mechanics, Wiley, New York. ZHU, D. Y., and Qian, Q. (2000). “Determination of Passive Earth Pressure Coefficient by the Method of Triangular Slices,” Canadian Geotechnical Journal, Vol. 37, No. 2, pp. 485–491. 8 8.1 Retaining Walls Introduction In Chapter 7, you were introduced to various theories of lateral earth pressure. Those theories will be used in this chapter to design various types of retaining walls. In general, retaining walls can be divided into two major categories: (a) conventional retaining walls and (b) mechanically stabilized earth walls. Conventional retaining walls can generally be classified into four varieties: 1. 2. 3. 4. Gravity retaining walls Semigravity retaining walls Cantilever retaining walls Counterfort retaining walls Gravity retaining walls (Figure 8.1a) are constructed with plain concrete or stone masonry. They depend for stability on their own weight and any soil resting on the masonry. This type of construction is not economical for high walls. In many cases, a small amount of steel may be used for the construction of gravity walls, thereby minimizing the size of wall sections. Such walls are generally referred to as semigravity walls (Figure 8.1b). Cantilever retaining walls (Figure 8.1c) are made of reinforced concrete that consists of a thin stem and a base slab. This type of wall is economical to a height of about 8 m. Figure 8.2 shows a cantilever retaining wall under construction. Counterfort retaining walls (Figure 8.1d) are similar to cantilever walls. At regular intervals, however, they have thin vertical concrete slabs known as counterforts that tie the wall and the base slab together. The purpose of the counterforts is to reduce the shear and the bending moments. To design retaining walls properly, an engineer must know the basic parameters— the unit weight, angle of friction, and cohesion—of the soil retained behind the wall and the soil below the base slab. Knowing the properties of the soil behind the wall enables the engineer to determine the lateral pressure distribution that has to be designed for. 375 376 Chapter 8: Retaining Walls Reinforcement Reinforcement Plain concrete or stone masonry (a) Gravity wall (b) Semigravity wall (c) Cantilever wall Counterfort (d) Counterfort wall Figure 8.1 Types of retaining wall There are two phases in the design of a conventional retaining wall. First, with the lateral earth pressure known, the structure as a whole is checked for stability. The structure is examined for possible overturning, sliding, and bearing capacity failures. Second, each component of the structure is checked for strength, and the steel reinforcement of each component is determined. This chapter presents the procedures for determining the stability of the retaining wall. Checks for strength can be found in any textbook on reinforced concrete. Some retaining walls have their backfills stabilized mechanically by including reinforcing elements such as metal strips, bars, welded wire mats, geotextiles, and 8.2 Proportioning Retaining Walls 377 Figure 8.2 A cantilever retaining wall under construction (Courtesy of Dharma Shakya, Geotechnical Solutions, Inc., Irvine, California) geogrids. These walls are relatively flexible and can sustain large horizontal and vertical displacements without much damage. Gravity and Cantilever Walls 8.2 Proportioning Retaining Walls In designing retaining walls, an engineer must assume some of their dimensions. Called proportioning, such assumptions allow the engineer to check trial sections of the walls for stability. If the stability checks yield undesirable results, the sections can be changed and rechecked. Figure 8.3 shows the general proportions of various retaining-wall components that can be used for initial checks. Note that the top of the stem of any retaining wall should not be less than about 0.3 m for proper placement of concrete. The depth, D, to the bottom of the base slab should be a minimum of 0.6 m. However, the bottom of the base slab should be positioned below the seasonal frost line. For counterfort retaining walls, the general proportion of the stem and the base slab is the same as for cantilever walls. However, the counterfort slabs may be about 0.3 m thick and spaced at center-to-center distances of 0.3H to 0.7H. 378 Chapter 8: Retaining Walls 0.3 m min 0.3 m min min 0.02 min 0.02 I I H H Stem 0.1 H D 0.12 to Heel 0.17 H Toe 0.12 to 0.17 H D 0.1 H 0.1 H 0.5 to 0.7 H 0.5 to 0.7 H (a) (b) Figure 8.3 Approximate dimensions for various components of retaining wall for initial stability checks: (a) gravity wall; (b) cantilever wall 8.3 Application of Lateral Earth Pressure Theories to Design The fundamental theories for calculating lateral earth pressure were presented in Chapter 7. To use these theories in design, an engineer must make several simple assumptions. In the case of cantilever walls, the use of the Rankine earth pressure theory for stability checks involves drawing a vertical line AB through point A, located at the edge of the heel of the base slab in Figure 8.4a. The Rankine active condition is assumed to exist along the vertical plane AB. Rankine active earth pressure equations may then be used to calculate the lateral pressure on the face AB of the wall. In the analysis of the wall’s stability, the force Pa(Rankine) , the weight of soil above the heel, and the weight Wc of the concrete all should be taken into consideration. The assumption for the development of Rankine active pressure along the soil face AB is theoretically correct if the shear zone bounded by the line AC is not obstructed by the stem of the wall. The angle, h, that the line AC makes with the vertical is h 5 45 1 fr 1 sin a a 2 2 sin21 ¢ ≤ 2 2 2 sin fr (8.1) 8.3 Application of Lateral Earth Pressure Theories to Design 379 A similar type of analysis may be used for gravity walls, as shown in Figure 8.4b. However, Coulomb’s active earth pressure theory also may be used, as shown in Figure 8.4c. If it is used, the only forces to be considered are Pa(Coulomb) and the weight of the wall, Wc . ␣ B C ␥1 1 c1 0 H Pa(Rankine) Ws ␥2 2 c2 H/3 Wc A (a) B ␥1 1 c1 0 Pa(Rankine) Ws ␥2 2 c2 Wc A (b) Figure 8.4 Assumption for the determination of lateral earth pressure: (a) cantilever wall; (b) and (c) gravity wall 380 Chapter 8: Retaining Walls ␥1 1 c1 Pa(Coulomb) ␦ Wc ␥2 2 c2 A (c) Figure 8.4 (continued) If Coulomb’s theory is used, it will be necessary to know the range of the wall friction angle d9 with various types of backfill material. Following are some ranges of wall friction angle for masonry or mass concrete walls: Backfill material Gravel Coarse sand Fine sand Stiff clay Silty clay Range of d9 (deg) 27–30 20–28 15–25 15–20 12–16 In the case of ordinary retaining walls, water table problems and hence hydrostatic pressure are not encountered. Facilities for drainage from the soils that are retained are always provided. 8.4 Stability of Retaining Walls A retaining wall may fail in any of the following ways: • • • • • It may overturn about its toe. (See Figure 8.5a.) It may slide along its base. (See Figure 8.5b.) It may fail due to the loss of bearing capacity of the soil supporting the base. (See Figure 8.5c.) It may undergo deep-seated shear failure. (See Figure 8.5d.) It may go through excessive settlement. The checks for stability against overturning, sliding, and bearing capacity failure will be described in Sections 8.5, 8.6, and 8.7. The principles used to estimate settlement 8.4 Stability of Retaining Walls (a) 381 (b) (c) Figure 8.5 Failure of retaining wall: (a) by overturning; (b) by sliding; (c) by bearing capacity failure; (d) by deep-seated shear failure (d) Angle ␣ with horizontal f a O For ␣ 10 d c e b Weak soil Figure 8.6 Deep-seated shear failure were covered in Chapter 5 and will not be discussed further. When a weak soil layer is located at a shallow depth—that is, within a depth of 1.5 times the width of the base slab of the retaining wall—the possibility of excessive settlement should be considered. In some cases, the use of lightweight backfill material behind the retaining wall may solve the problem. 382 Chapter 8: Retaining Walls Deep shear failure can occur along a cylindrical surface, such as abc shown in Figure 8.6, as a result of the existence of a weak layer of soil underneath the wall at a depth of about 1.5 times the width of the base slab of the retaining wall. In such cases, the critical cylindrical failure surface abc has to be determined by trial and error, using various centers such as O. The failure surface along which the minimum factor of safety is obtained is the critical surface of sliding. For the backfill slope with a less than about 10°, the critical failure circle apparently passes through the edge of the heel slab (such as def in the figure). In this situation, the minimum factor of safety also has to be determined by trial and error by changing the center of the trial circle. 8.5 Check for Overturning Figure 8.7 shows the forces acting on a cantilever and a gravity retaining wall, based on the assumption that the Rankine active pressure is acting along a vertical plane AB drawn through the heel of the structure. Pp is the Rankine passive pressure; recall that its magnitude is [from Eq. (7.63)]. Pp 5 12Kpg2D2 1 2c2r "KpD where g2 5 unit weight of soil in front of the heel and under the base slab Kp 5 Rankine passive earth pressure coefficient 5 tan2 (45 1 f2r >2) c2r , f2r 5 cohesion and effective soil friction angle, respectively The factor of safety against overturning about the toe—that is, about point C in Figure 8.7—may be expressed as FS(overturning) 5 SMR SMo (8.2) where SMo 5 sum of the moments of forces tending to overturn about point C SMR 5 sum of the moments of forces tending to resist overturning about point C The overturning moment is SMo 5 Ph ¢ Hr ≤ 3 (8.3) where Ph 5 Pa cos a . To calculate the resisting moment, SMR (neglecting Pp ), a table such as Table 8.1 can be prepared. The weight of the soil above the heel and the weight of the concrete (or masonry) are both forces that contribute to the resisting moment. Note that the force Pv also contributes to the resisting moment. Pv is the vertical component of the active force Pa , or Pv 5 Pa sin a The moment of the force Pv about C is Mv 5 PvB 5 Pa sin aB (8.4) 8.5 Check for Overturning 2 ␣ 383 A ␥1 1 c1 0 1 H P Pa 3 4 D Ph Pp 5 B qheel C ␥2 2 c2 qtoe B 2 ␣ A ␥1 1 c1 0 1 P Pa Ph 3 4 5 D Pp 6 B qheel C ␥2 2 c2 qtoe B Figure 8.7 Check for overturning, assuming that the Rankine pressure is valid where B 5 width of the base slab. Once S MR is known, the factor of safety can be calculated as FS(overturning) 5 M1 1 M2 1 M3 1 M4 1 M5 1 M6 1 Mv Pa cos a(Hr>3) (8.5) The usual minimum desirable value of the factor of safety with respect to overturning is 2 to 3. 384 Chapter 8: Retaining Walls Table 8.1 Procedure for Calculating SMR Section (1) Area (2) 1 2 3 4 5 6 A1 A2 A3 A4 A5 A6 Weight/unit length of wall (3) Moment arm measured from C (4) W1 5 g1 3 A 1 W2 5 g1 3 A 2 W3 5 gc 3 A 3 W4 5 gc 3 A 4 W5 5 gc 3 A 5 W6 5 gc 3 A 6 Pv SV X1 X2 X3 X4 X5 X6 B Moment about C (5) M1 M2 M3 M4 M5 M6 Mv S MR (Note: gl 5 unit weight of backfill gc 5 unit weight of concrete) Some designers prefer to determine the factor of safety against overturning with the formula M1 1 M2 1 M3 1 M4 1 M5 1 M6 FS(overturning) 5 (8.6) Pa cos a(Hr>3) 2 Mv 8.6 Check for Sliding along the Base The factor of safety against sliding may be expressed by the equation FS(sliding) 5 S FRr S Fd (8.7) where S FRr 5 sum of the horizontal resisting forces S Fd 5 sum of the horizontal driving forces Figure 8.8 indicates that the shear strength of the soil immediately below the base slab may be represented as s 5 sr tan dr 1 car where dr 5 angle of friction between the soil and the base slab car 5 adhesion between the soil and the base slab Thus, the maximum resisting force that can be derived from the soil per unit length of the wall along the bottom of the base slab is Rr 5 s(area of cross section) 5 s(B 3 1) 5 Bsr tan dr 1 Bcar However, Bsr 5 sum of the vertical force 5 S V(see Table 8.1) so Rr 5 (S V) tan dr 1 Bcar 385 8.6 Check for Sliding along the Base ␥1 1 c1 V Ph D Pp ␥2 2 c2 B Figure 8.8 Check for sliding along the base Figure 8.8 shows that the passive force Pp is also a horizontal resisting force. Hence, S FRr 5 (S V) tan dr 1 Bcar 1 Pp (8.8) The only horizontal force that will tend to cause the wall to slide (a driving force) is the horizontal component of the active force Pa , so S Fd 5 Pa cos a (8.9) Combining Eqs. (8.7), (8.8), and (8.9) yields FS(sliding) 5 (S V) tan dr 1 Bcra 1 Pp Pa cos a (8.10) A minimum factor of safety of 1.5 against sliding is generally required. In many cases, the passive force Pp is ignored in calculating the factor of safety with respect to sliding. In general, we can write dr 5 k1f2r and car 5 k2c2r . In most cases, k1 and k2 are in the range from 12 to 23 . Thus, FS(sliding) 5 (S V) tan (k1f2r ) 1 Bk2c2r 1 Pp Pa cos a (8.11) If the desired value of FS(sliding) is not achieved, several alternatives may be investigated (see Figure 8.9): • Increase the width of the base slab (i.e., the heel of the footing). • Use a key to the base slab. If a key is included, the passive force per unit length of the wall becomes Pp 5 where Kp 5 tan2 ¢ 45 1 1 g D2K 1 2c2r D1"Kp 2 2 1 p f2r ≤. 2 386 Chapter 8: Retaining Walls Use of a dead man anchor ␥1 1 c1 D D1 Base slab increase Use of a base key Pp ␥2 2 c2 Figure 8.9 Alternatives for increasing the factor of safety with respect to sliding • Use a deadman anchor at the stem of the retaining wall. • Another possible way to increase the value of FS(sliding) is to consider reducing the value of Pa [see Eq. (8.11)]. One possible way to do so is to use the method developed by Elman and Terry (1988). The discussion here is limited to the case in which the retaining wall has a horizontal granular backfill (Figure 8.10). In Figure 8.10, the active force, Pa, is horizontal (␣ ⫽ 0) so that Pacos ␣ ⫽ Ph ⫽ Pa and Pa sin ␣ ⫽ Pv ⫽ 0 However, Pa ⫽ Pa(1) ⫹ Pa(2) (8.12) The magnitude of Pa(2) can be reduced if the heel of the retaining wall is sloped as shown in Figure 8.10. For this case, Pa ⫽ Pa(1) ⫹ APa(2) (8.13) The magnitude of A, as shown in Table 8.2, is valid for ␣⬘ ⫽ 45°. However note that in Figure 8.10a Pa(1) 5 1 g K (Hr 2 Dr ) 2 2 1 a and Pa 5 1 g K Hr2 2 1 a Hence, Pa(2) 5 1 g K 3Hr2 2 (H 9 2 D 9 ) 24 2 1 a 387 8.7 Check for Bearing Capacity Failure ␥1 1 c1= 0 ␥1 1 c1= 0 H Pa = [Eq. (8.12)] Pa(1) D D Pa(1) Pa = [Eq. (8.13)] APa(2) Pa(2) α (a) (b) Figure 8.10 Retaining wall with sloped heel Table 8.2 Variation of A with 1⬘ (for ␣⬘ ⫽ 45°) Soil friction angle, 1ⴕ (deg) A 20 25 30 35 40 0.28 0.14 0.06 0.03 0.018 So, for the active pressure diagram shown in Figure 8.10b, Pa 5 1 A g1Ka (H 9 2 D 9 ) 2 1 g1Ka 3H 92 2 (H 9 2 D 9 ) 24 2 2 (8.14) Sloping the heel of a retaining wall can thus be extremely helpful in some cases. 8.7 Check for Bearing Capacity Failure The vertical pressure transmitted to the soil by the base slab of the retaining wall should be checked against the ultimate bearing capacity of the soil. The nature of variation of the vertical pressure transmitted by the base slab into the soil is shown in Figure 8.11. Note that qtoe and qheel are the maximum and the minimum pressures occurring at the ends of the toe and heel sections, respectively. The magnitudes of qtoe and qheel can be determined in the following manner: The sum of the vertical forces acting on the base slab is S V (see column 3 of Table 8.1), and the horizontal force Ph is Pa cos a. Let R 5 S V 1 Ph (8.15) be the resultant force. The net moment of these forces about point C in Figure 8.11 is Mnet 5 SMR 2 SMo (8.16) 388 Chapter 8: Retaining Walls ␥1 1 c1 0 V V R Ph Pa cos ␣ X ␥2 2 c2 D Ph Ph C E qmax qtoe qmin qheel e B/2 y B/2 Figure 8.11 Check for bearing capacity failure Note that the values of SMR and SMo were previously determined. [See Column 5 of Table 8.1 and Eq. (8.3)]. Let the line of action of the resultant R intersect the base slab at E. Then the distance CE 5 X 5 Mnet SV (8.17) Hence, the eccentricity of the resultant R may be expressed as e5 B 2 CE 2 (8.18) The pressure distribution under the base slab may be determined by using simple principles from the mechanics of materials. First, we have q5 Mnety SV 6 A I where Mnet 5 moment 5 (SV)e I 5 moment of inertia per unit length of the base section 5 121 (1) (B3 ) (8.19) 8.7 Check for Bearing Capacity Failure 389 For maximum and minimum pressures, the value of y in Eq. (8.19) equals B>2. Substituting into Eq. (8.19) gives qmax 5 qtoe 5 SV 1 (B) (1) e(SV) B 2 1 ¢ ≤ (B3 ) 12 5 SV 6e ¢1 1 ≤ B B (8.20) Similarly, qmin 5 qheel 5 SV 6e ¢1 2 ≤ B B (8.21) Note that SV includes the weight of the soil, as shown in Table 8.1, and that when the value of the eccentricity e becomes greater than B>6, qmin [Eq. (8.21)] becomes negative. Thus, there will be some tensile stress at the end of the heel section. This stress is not desirable, because the tensile strength of soil is very small. If the analysis of a design shows that e . B>6, the design should be reproportioned and calculations redone. The relationships pertaining to the ultimate bearing capacity of a shallow foundation were discussed in Chapter 3. Recall that [Eq. (3.40)]. qu 5 c2r NcFcdFci 1 qNqFqdFqi 1 12g2BrNgFgdFgi (8.22 ) where q 5 g2D Br 5 B 2 2e 1 2 Fqd Fcd 5 Fqd 2 Nc tan f2r Fqd 5 1 1 2 tan f2r (1 2 sin f2r ) 2 D Br Fgd 5 1 Fci 5 Fqi 5 ¢ 1 2 Fgi 5 ¢1 2 c° 2 ≤ 90° c° 2 ≤ f2r ° c° 5 tan21 ¢ Pa cos a ≤ SV Note that the shape factors Fcs , Fqs , and Fgs given in Chapter 3 are all equal to unity, because they can be treated as a continuous foundation. For this reason, the shape factors are not shown in Eq. (8.22). Once the ultimate bearing capacity of the soil has been calculated by using Eq. (8.22), the factor of safety against bearing capacity failure can be determined: FS(bearing capacity) 5 qu qmax (8.23) Generally, a factor of safety of 3 is required. In Chapter 3, we noted that the ultimate bearing capacity of shallow foundations occurs at a settlement of about 10% of the foundation width. 390 Chapter 8: Retaining Walls In the case of retaining walls, the width B is large. Hence, the ultimate load qu will occur at a fairly large foundation settlement. A factor of safety of 3 against bearing capacity failure may not ensure that settlement of the structure will be within the tolerable limit in all cases. Thus, this situation needs further investigation. An alternate relationship to Eq. (8.22) will be Eq. (3.67), or qu 5 crNc(ei)Fcd 1 qNq(ei)Fqd 1 12g2BNg(ei)Fgd Since Fgd 5 1, qu 5 crNc(ei)Fcd 1 qNq(ei)Fqd 1 12g2BNg(ei) (8.24) The bearing capacity factors, Nc(ei), Nq(ei), and Ng(ei) were given in Figures 3.26 through 3.28. Example 8.1 The cross section of a cantilever retaining wall is shown in Figure 8.12. Calculate the factors of safety with respect to overturning, sliding, and bearing capacity. 10 0.5 m H1 = 0.458 m 5 1 = 18 kN/m3 1= 30 c1= 0 H2 = 6 m 1 Pa Pv 10 4 Ph 2 1.5 m = D 0.7 m 3 H3 = 0.7 m C 0.7 m 0.7 m 2.6 m Figure 8.12 Calculation of stability of a retaining wall ␥2 = 19 kN/m3 2 = 20 c2 = 40 kN/m2 8.7 Check for Bearing Capacity Failure 391 Solution From the figure, H⬘ ⫽ H1 ⫹ H2 ⫹ H3 ⫽ 2.6 tan 10° ⫹ 6 ⫹ 0.7 ⫽ 0.458 ⫹ 6 ⫹ 0.7 ⫽ 7.158 m The Rankine active force per unit length of wall ⫽ Pp 5 12g1Hr2Ka . For 1⬘ ⫽ 30° and ␣ ⫽ 10°, Ka is equal to 0.3532. (See Table 7.1.) Thus, Pa 5 12 (18) (7.158) 2 (0.3532) 5 162.9 kN>m Pv ⫽ Pa sin10° ⫽ 162.9 (sin10°) ⫽ 28.29 kN/m and Ph ⫽ Pa cos10° ⫽ 162.9 (cos10°) ⫽ 160.43 kN/m Factor of Safety against Overturning The following table can now be prepared for determining the resisting moment: Section no.a Area (m2) Weight/unit length (kN/m) Moment arm from point C (m) 70.74 14.15 1.15 0.833 66.02 280.80 10.71 Pv ⫽ 28.29 ⌺V ⫽ 470.71 2.0 2.7 3.13 4.0 6 ⫻ 0.5 ⫽ 3 5 0.6 1 2 1 2 (0.2)6 3 4 5 4 ⫻ 0.7 ⫽ 2.8 6 ⫻ 2.6 ⫽ 15.6 1 2 (2.6) (0.458) 5 0.595 Moment (kN-m/m) 81.35 11.79 132.04 758.16 33.52 113.16 1130.02 ⫽ ⌺MR a For section numbers, refer to Figure 8.12 ␥concrete ⫽ 23.58 kN/m3 The overturning moment Mo 5 Ph a 7.158 H9 b 5 160.43a b 5 382.79 kN-m>m 3 3 and FS (overturning) 5 SMR 1130.02 5 5 2.95 . 2, OK Mo 382.79 Factor of Safety against Sliding From Eq. (8.11), FS (sliding) 5 (SV)tan(k1fr2 ) 1 Bk2cr2 1 Pp Pacosa 392 Chapter 8: Retaining Walls Let k1 5 k2 5 32 . Also, Pp 5 12Kpg2D2 1 2cr2 !KpD Kp 5 tan2 a45 1 fr2 b 5 tan2 (45 1 10) 5 2.04 2 and D ⫽ 1.5 m So Pp 5 12 (2.04) (19) (1.5) 2 1 2(40) ( !2.04) (1.5) ⫽ 43.61 ⫹ 171.39 ⫽ 215 kN/m Hence, (470.71)tana FS (sliding) 5 5 2 3 20 2 b 1 (4) a b (40) 1 215 3 3 160.43 111.56 1 106.67 1 215 5 2.7 > 1.5, OK 160.43 Note: For some designs, the depth D in a passive pressure calculation may be taken to be equal to the thickness of the base slab. Factor of Safety against Bearing Capacity Failure Combining Eqs. (8.16), (8.17), and (8.18) yields SMR 2 SMo B 4 1130.02 2 382.79 2 5 2 2 SV 2 470.71 B 4 5 0.411 m , 5 5 0.666 m 6 6 e5 Again, from Eqs. (8.20) and (8.21) q toe heel 5 SV 6e 470.71 6 3 0.411 a1 6 b 5 a1 6 b 5 190.2 kN>m2 (toe) B B 4 4 5 45.13 kN>m2 (heel) The ultimate bearing capacity of the soil can be determined from Eq. (8.22) qu 5 c29NcFcdFci 1 qNqFqdFqi 1 1 g B9NgFgdFgi 2 2 8.7 Check for Bearing Capacity Failure 393 For 2⬘ ⫽ 20° (see Table 3.3), Nc ⫽ 14.83, Nq ⫽ 6.4, and N␥ ⫽ 5.39. Also, q ⫽ ␥2D ⫽ (19) (1.5) ⫽ 28.5 kN/m2 B⬘ ⫽ B ⫺ 2e ⫽ 4 ⫺ 2(0.411) ⫽ 3.178 m 1 2 Fqd 1 2 1.148 Fcd 5 Fqd 2 5 1.148 2 5 1.175 Nctanfr2 (14.83) (tan 20) 2 Fqd 5 1 1 2 tanfr2 (1 2 sinfr) 2 a D 1.5 b 5 1 1 0.315a b 5 1.148 Br 3.178 F␥d ⫽ 1 Fci 5 Fqi 5 a1 2 c° 2 b 90° and c 5 tan21 a Pacosa 160.43 b 5 tan21 a b 5 18.82° SV 470.71 So Fci 5 Fqi 5 a1 2 18.82 2 b 5 0.626 90 and Fgi 5 a1 2 c 2 18.82 2 b 5 a1 2 b <0 f2r 20 Hence, qu 5 (40) (14.83) (1.175) (0.626) 1 (28.5) (6.4) (1.148) (0.626) 1 12 (19) (5.93) (3.178) (1) (0) ⫽ 436.33 ⫹ 131.08 ⫹ 0 ⫽ 567.41 kN/m2 and FS (bearing capacity) 5 qu 567.41 5 5 2.98 q toe 190.2 Note: FS(bearing capacity) is less than 3. Some repropertioning will be needed. ■ Example 8.2 A gravity retaining wall is shown in Figure 8.13. Use dr 5 2>3f1r and Coulomb’s active earth pressure theory. Determine 394 Chapter 8: Retaining Walls ␥1 18.5 kN/m3 1 32° c1 0 P 5.7 m 5m 2 Pa 1 ␦ 15° 2.83 m Ph 3 75° 2.167 m 1.5 m 0.27 m 0.6 m 0.8 m 1.53 m 4 C 0.3 m 0.8 m 3.5 m ␥2 18 kN/m3 2 24° c2 30 kN/m2 Figure 8.13 Gravity retaining wall (not to scale) a. The factor of safety against overturning b. The factor of safety against sliding c. The pressure on the soil at the toe and heel Solution The height Hr 5 5 1 1.5 5 6.5 m Coulomb’s active force is Pa 5 12 g1Hr2Ka With a 5 0°, b 5 75°, dr 5 2>3f1r , and f1r 5 32°, Ka 5 0.4023. (See Table 7.4.) So, Pa 5 12 (18.5) (6.5) 2 (0.4023) 5 157.22 kN>m Ph 5 Pa cos (15 1 23f1r ) 5 157.22 cos 36.33 5 126.65 kN>m and Pv 5 Pa sin (15 1 23f1r ) 5 157.22 sin 36.33 5 93.14 kN>m 8.7 Check for Bearing Capacity Failure 395 Part a: Factor of Safety against Overturning From Figure 8.13, one can prepare the following table: Area no. 1 2 3 4 * 1 2 (5.7) Moment arm from C (m) Weight* (kN/m) Area (m2 ) (1.53) 5 4.36 (0.6) (5.7) 5 3.42 1 2 (0.27) (5.7) 5 0.77 < (3.5) (0.8) 5 2.8 102.81 80.64 18.16 66.02 Pv 5 93.14 SV 5 360.77 kN>m 2.18 1.37 0.98 1.75 2.83 Moment (kN-m/m) 224.13 110.48 17.80 115.54 263.59 SMR 5 731.54 kN-m>m gconcrete 5 23.58 kN>m3 Note that the weight of the soil above the back face of the wall is not taken into account in the preceding table. We have Overturning moment 5 Mo 5 Ph ¢ Hr ≤ 5 126.65(2.167) 5 274.45 kN-m>m 3 Hence, FS(overturning) 5 SMR 731.54 5 5 2.67 + 2, OK SMo 274.45 Part b: Factor of Safety against Sliding We have FS(sliding) 5 2 2 (SV) tan ¢ f2r ≤ 1 c2r B 1 Pp 3 3 Ph Pp 5 12Kpg2D2 1 2c2r "KpD and Kp 5 tan2 ¢45 1 24 ≤ 5 2.37 2 Hence, Pp 5 12 (2.37) (18) (1.5) 2 1 2(30) (1.54) (1.5) 5 186.59 kN>m So FS(sliding) 5 2 2 360.77 tan ¢ 3 24≤ 1 (30) (3.5) 1 186.59 3 3 126.65 396 Chapter 8: Retaining Walls 5 103.45 1 70 1 186.59 5 2.84 126.65 If Pp is ignored, the factor of safety is 1.37. Part c: Pressure on Soil at Toe and Heel From Eqs. (8.16), (8.17), and (8.18), e5 qtoe 5 SMR 2 SMo B 3.5 731.54 2 274.45 B 2 5 2 5 0.483 , 5 0.583 2 SV 2 360.77 6 (6) (0.483) SV 6e 360.77 B1 1 R 5 B1 1 R 5 188.43 kN , m2 B B 3.5 3.5 and qheel 5 8.8 (6) (0.483) V 6e 360.77 B1 2 R 5 B1 2 R 5 17.73 kN , m2 B B 3.5 3.5 ■ Construction Joints and Drainage from Backfill Construction Joints A retaining wall may be constructed with one or more of the following joints: 1. Construction joints (see Figure 8.14a) are vertical and horizontal joints that are placed between two successive pours of concrete. To increase the shear at the joints, keys may be used. If keys are not used, the surface of the first pour is cleaned and roughened before the next pour of concrete. 2. Contraction joints (Figure 8.14b) are vertical joints (grooves) placed in the face of a wall (from the top of the base slab to the top of the wall) that allow the concrete to shrink without noticeable harm. The grooves may be about 6 to 8 mm wide and 12 to 16 mm deep. 3. Expansion joints (Figure 8.14c) allow for the expansion of concrete caused by temperature changes; vertical expansion joints from the base to the top of the wall may also be used. These joints may be filled with flexible joint fillers. In most cases, horizontal reinforcing steel bars running across the stem are continuous through all joints. The steel is greased to allow the concrete to expand. Drainage from the Backfill As the result of rainfall or other wet conditions, the backfill material for a retaining wall may become saturated, thereby increasing the pressure on the wall and perhaps creating an unstable condition. For this reason, adequate drainage must be provided by means of weep holes or perforated drainage pipes. (See Figure 8.15.) When provided, weep holes should have a minimum diameter of about 0.1 m and be adequately spaced. Note that there is always a possibility that backfill material may 8.8 Construction Joints and Drainage from Backfill 397 Roughened surface Keys (a) Back of wall Back of wall Contraction joint Face of wall (b) Expansion joint (c) Face of wall Figure 8.14 (a) Construction joints; (b) contraction joint; (c) expansion joint Weep hole Filter material Filter material Perforated pipe (a) (b) Figure 8.15 Drainage provisions for the backfill of a retaining wall: (a) by weep holes; (b) by a perforated drainage pipe be washed into weep holes or drainage pipes and ultimately clog them. Thus, a filter material needs to be placed behind the weep holes or around the drainage pipes, as the case may be; geotextiles now serve that purpose. Two main factors influence the choice of filter material: The grain-size distribution of the materials should be such that (a) the soil to be protected is not washed into the filter and (b) excessive hydrostatic pressure head is not created in the soil with a lower 398 Chapter 8: Retaining Walls hydraulic conductivity (in this case, the backfill material). The preceding conditions can be satisfied if the following requirements are met (Terzaghi and Peck, 1967): D15(F) D85(B) D15(F) D15(B) ,5 3to satisfy condition(a)4 (8.25) .4 3to satisfy condition(b) 4 (8.26) In these relations, the subscripts F and B refer to the filter and the base material (i.e., the backfill soil), respectively. Also, D15 and D85 refer to the diameters through which 15% and 85% of the soil (filter or base, as the case may be) will pass. Example 8.3 gives the procedure for designing a filter. Example 8.3 Figure 8.16 shows the grain-size distribution of a backfill material. Using the conditions outlined in Section 8.8, determine the range of the grain-size distribution for the filter material. 100 Range for filter material Percent finer 80 D85(B) Backfill material 60 D50(B) 25 D50(B) 40 20 5 D85(B) 4 D15(B) D15(B) 20 D15(B) 0 10 5 2 1 0.5 0.2 0.1 Grain size (mm) 0.05 0.02 0.01 Figure 8.16 Determination of grain-size distribution of filter material Solution From the grain-size distribution curve given in the figure, the following values can be determined: D15(B) 5 0.04 mm D85(B) 5 0.25 mm D50(B) 5 0.13 mm 8.9 Gravity Retaining-Wall Design for Earthquake Conditions 399 Conditions of Filter 1. 2. 3. 4. D15(F) D15(F) D50(F) D15(F) should be less than 5D85(B) ; that is, 5 3 0.25 5 1.25 mm. should be greater than 4D15(B) ; that is, 4 3 0.04 5 0.16 mm. should be less than 25D50(B) ; that is, 25 3 0.13 5 3.25 mm. should be less than 20D15(B) ; that is, 20 3 0.04 5 0.8 mm. These limiting points are plotted in Figure 8.16. Through them, two curves can be drawn that are similar in nature to the grain-size distribution curve of the backfill material. These curves define the range of the filter material to be used. ■ 8.9 Gravity Retaining-Wall Design for Earthquake Conditions Even in mild earthquakes, most retaining walls undergo limited lateral displacement. Richards and Elms (1979) proposed a procedure for designing gravity retaining walls for earthquake conditions that allows limited lateral displacement. This procedure takes into consideration the wall inertia effect. Figure 8.17 shows a retaining wall with various forces acting on it, which are as follows (per unit length of the wall): a. Ww ⫽ weight of the wall b. Pae ⫽ active force with earthquake condition taken into consideration (Section 7.7) The backfill of the wall and the soil on which the wall is resting are assumed cohesionless. Considering the equilibrium of the wall, it can be shown that Ww 5 C12g1H 2 (1 2 kv )KaeD CIE (8.27) a ␥1 1 c1 = 0 Pae ␦ 90 ⫺  kvWw H khWw Ww S = khWw Pae sin( N = Ww ␦⬘) kvWw Pae cos( ␦⬘) z  ␥2 2 c2= 0 Figure 8.17 Stability of a retaining wall under earthquake forces 400 Chapter 8: Retaining Walls where ␥1 ⫽ unit weight of the backfill; CIE 5 and u 9 5 tan21 a sin(b 2 d9 ) 2 cos(b 2 d9 )tanf29 (1 2 kv ) (tanf29 2 tan u9) (8.28) kk b 1 2 kv For a detailed derivation of Eq. (8.28), see Das (1983). Based on Eqs. (8.27) and (8.28), the following procedure may be used to determine the weight of the retaining wall, Ww, for tolerable displacement that may take place during an earthquake. 1. Determine the tolerable displacement of the wall, ⌬. 2. Obtain a design value of kk from kk 5 A a a 0.2A 2v 0.25 b AaD (8.29) In Eq. (8.29), A and Aa are effective acceleration coefficients and ⌬ is displacement in inches. The magnitudes of Aa and Av are given by the Applied Technology Council (1978) for various regions of the United States 3. Assume that kv ⫽ 0, and, with the value of kk obtained, calculate Kae from Eq. (7.43). 4. Use the value of Kae determined in Step 3 to obtain the weight of the wall (Ww) 5. Apply a factor of safety to the value of Ww obtained in Step 4. Example 8.4 Refer to Figure 8.18. For kv ⫽ 0 and kk ⫽ 0.3, determine: a. Weight of the wall for static condition b. Weight of the wall for zero displacement during an earthquake c. Weight of the wall for lateral displacement of 38 mm (1.5 in.) during an earthquake 1 = 36 ␥1 = 16 kN/m3 ␦ = 2/3 1 5m 2 = 36 ␥2 = 16 kN/m3 Figure 8.18 8.9 Gravity Retaining-Wall Design for Earthquake Conditions 401 For part c, assume that Aa ⫽ 0.2 and Av ⫽ 0.2. For parts a, b, and c, use a factor of safety of 1.5. Solution Part a For static conditions, ⬘ ⫽ 0 and Eq. (8.28) becomes CIE 5 sin(b 2 dr ) 2 cos(b 2 dr )tanf2r tanf2r For  ⫽ 90°, ␦⬘ ⫽ 24° and ⬘2 ⫽ 36°, CIE 5 sin(90 2 24) 2 cos(90 2 24)tan 36 5 0.85 tan 36 For static conditions, Kae ⫽ Ka, so 1 Ww 5 gH 2KaCIE 2 For Ka ⬇ 0.2349 [Table 7.4], 1 Ww 5 (16) (5) 2 (0.2349) (0.85) 5 39.9 kN>m 2 With a factor of safety of 1.5, Ww ⫽ (39.9)(1.5) ⫽ 59.9 kN/m Part b For zero displacement, kv ⫽ 0, CIE 5 tan ur 5 CIE 5 sin(b 2 dr ) 2 cos(b 2 dr )tan f2r tan f2r 2 tan ur kh 0.3 5 5 0.3 1 2 kv 120 sin(90 2 24) 2 cos(90 2 24)tan 36 5 1.45 tan 36 2 0.3 For kh ⫽ 0.3, 1⬘ ⫽ 36° and ␦⬘ ⫽ 21⬘/3, the value of Kae ⬇ 0.48 (Table 7.6). Ww 5 12g1H 2 (1 2 kv )KaeCIE 5 12 (16) (5) 2 (1 2 0) (0.48) (1.45) 5 139.2 kN>m With a factor of safety of 1.5, Ww ⫽ 208.8 kN/m Part c For a lateral displacement of 38 mm, kh 5 A a a 0.2A 2v 0.25 (0.2) (0.2) 2 0.25 b 5 (0.2) c d 5 0.081 AaD (0.2) (38>25.4) 402 Chapter 8: Retaining Walls tan ur 5 CIE 5 kh 0.081 5 5 0.081 1 2 kv 120 sin (90 2 24) 2 cos (90 2 24)tan 36 5 0.957 tan 36 2 0.081 1 g H 2KaeClE 2 1 h Ww 5 ⬇ 0.29 [Table 7.6] 1 Ww 5 (16) (5) 2 (0.29) (0.957) 5 55.5 kN>m 2 With a factor of safety of 1.5, Ww ⫽ 83.3 kN/m 8.10 ■ Comments on Design of Retaining Walls and a Case Study In Section 8.3, it was suggested that the active earth pressure coefficient be used to estimate the lateral force on a retaining wall due to the backfill. It is important to recognize the fact that the active state of the backfill can be established only if the wall yields sufficiently, which does not happen in all cases. The degree to which the wall yields depends on its height and the section modulus. Furthermore, the lateral force of the backfill depends on several factors identified by Casagrande (1973): 1. 2. 3. 4. 5. 6. 7. Effect of temperature Groundwater fluctuation Readjustment of the soil particles due to creep and prolonged rainfall Tidal changes Heavy wave action Traffic vibration Earthquakes Insufficient wall yielding combined with other unforeseen factors may generate a larger lateral force on the retaining structure, compared with that obtained from the active earth-pressure theory. This is particularly true in the case of gravity retaining walls, bridge abutments, and other heavy structures that have a large section modulus. Case Study for the Performance of a Cantilever Retaining Wall Bentler and Labuz (2006) have reported the performance of a cantilever retaining wall built along Interstate 494 in Bloomington, Minnesota. The retaining wall had 83 panels, each having a length of 9.3 m. The panel height ranged from 4.0 m to 7.9 m. One of the 7.9 m high panels was instrumented with earth pressure cells, tiltmeters, strain gauges, and 8.10 Comments on Design of Retaining Walls and a Case Study 403 Granular backfill (SP) ␥1 = 18.9 kN/m3 1= 35 to 39 (Av. 37) 2.4 7.9 m Figure 8.19 Schematic diagram of the retaining wall (drawn to scale) Poorly graded sand, and sand and gravel inclinometer casings. Figure 8.19 shows a schematic diagram (cross section) of the wall panel. Some details on the backfill and the foundation material are: • Granular Backfill Effective size, D10 ⫽ 0.13 mm Uniformity coefficient, Cu ⫽ 3.23 Coefficient of gradation, Cc ⫽ 1.4 Unified soil classification ⫺ SP Compacted unit weight, ␥1 ⫽ 18.9 kN/m3 Triaxial friction angle, 1⬘ ⫺ 35° to 39° (average 37°) • Foundation Material Poorly graded sand and sand with gravel (medium dense to dense) The backfill and compaction of the granular material started on October 28, 2001 in stages and reached a height of 7.6 m on November 21, 2001. The final 0.3 m of soil was placed the following spring. During backfilling, the wall was continuously going through translation (see Section 7.9). Table 8.3 is a summary of the backfill height and horizontal translation of the wall. Table 8.3 Horizontal Translation with Backfill Height Day Backfill height (m) Horizontal translation (mm) 1 2 2 3 4 5 11 24 54 0.0 1.1 2.8 5.2 6.1 6.4 6.7 7.3 7.6 0 0 0 2 4 6 9 12 11 404 Chapter 8: Retaining Walls Height of fill above footing (m) 8 6.1 m 6 Observed pressure Rankine active pressure 4 (⬘1 ⬇ 37) 2 0 0 10 20 30 40 50 Lateral pressure (kN/m2) Figure 8.20 Observed lateral pressure distribution after fill height reached 6.1 m (Based on Bentler and Labuz, 2006) Figure 8.20 shows a typical plot of the variation of lateral earth pressure after compaction, a⬘, when the backfill height was 6.1 m (October 31, 2001) along with the plot of Rankine active earth pressure (1⬘ ⫽ 37°). Note that the measured lateral (horizontal) pressure is higher at most heights than that predicted by the Rankine active pressure theory, which may be due to residual lateral stresses caused by compaction. The measured lateral stress gradually reduced with time. This is demonstrated in Figure 8.21 which shows a plot of the variation of a⬘ with depth (November 27, 2001) when the height of the backfill was 7.6 m. The lateral pressure was lower at practically all depths compared to the Rankine active earth pressure. Another point of interest is the nature of variation of qmax and qmin (see Figure 8.11). As shown in Figure 8.11, if the wall rotates about C, qmax will be at the toe and qmin will be at the heel. However, for the case of the retaining wall under consideration (undergoing Height of fill above footing (m) 8 7.6 m 6 Rankine active pressure (1 ⬇ 37) 4 2 0 Observed pressure 0 10 20 30 40 Lateral pressure (kN/m2) 50 Figure 8.21 Observed pressure distribution on November 27, 2001 (Based on Bentler and Labuz, 2006) 8.11 Soil Reinforcement 405 horizontal translation), qmax was at the heel of the wall with qmin at the toe. On November 27, 2001, when the height of the fill was 7.6 m, qmax at the heel was about 140 kN/m2, which was approximately equal to (␥1)(height of fill) ⫽(18.9)(7.6) ⫽ 143.6 kN/m2. Also, at the toe, qmin was about 40 kN/m2, which suggests that the moment from lateral force had little effect on the vertical effective stress below the heel. The lessons learned from this case study are the following: a. Retaining walls may undergo lateral translation which will affect the variation of qmax and qmin along the base slab. b. Initial lateral stress caused by compaction gradually decreases with time and lateral movement of the wall. Mechanically Stabilized Retaining Walls More recently, soil reinforcement has been used in the construction and design of foundations, retaining walls, embankment slopes, and other structures. Depending on the type of construction, the reinforcements may be galvanized metal strips, geotextiles, geogrids, or geocomposites. Sections 8.11 and 8.12 provide a general overview of soil reinforcement and various reinforcement materials. Reinforcement materials such as metallic strips, geotextiles, and geogrids are now being used to reinforce the backfill of retaining walls, which are generally referred to as mechanically stabilized retaining walls. The general principles for designing these walls are given in the following sections. 8.11 Soil Reinforcement The use of reinforced earth is a recent development in the design and construction of foundations and earth-retaining structures. Reinforced earth is a construction material made from soil that has been strengthened by tensile elements such as metal rods or strips, nonbiodegradable fabrics (geotextiles), geogrids, and the like. The fundamental idea of reinforcing soil is not new; in fact, it goes back several centuries. However, the present concept of systematic analysis and design was developed by a French engineer, H. Vidal (1966). The French Road Research Laboratory has done extensive research on the applicability and the beneficial effects of the use of reinforced earth as a construction material. This research has been documented in detail by Darbin (1970), Schlosser and Long (1974), and Schlosser and Vidal (1969). The tests that were conducted involved the use of metallic strips as reinforcing material. Retaining walls with reinforced earth have been constructed around the world since Vidal began his work. The first reinforced-earth retaining wall with metal strips as reinforcement in the United States was constructed in 1972 in southern California. The beneficial effects of soil reinforcement derive from (a) the soil’s increased tensile strength and (b) the shear resistance developed from the friction at the soil-reinforcement interfaces. Such reinforcement is comparable to that of concrete structures. Currently, most reinforced-earth design is done with free-draining granular soil only. Thus, the effect of pore water development in cohesive soils, which, in turn, reduces the shear strength of the soil, is avoided. 406 Chapter 8: Retaining Walls 8.12 Considerations in Soil Reinforcement Metal Strips In most instances, galvanized steel strips are used as reinforcement in soil. However, galvanized steel is subject to corrosion. The rate of corrosion depends on several environmental factors. Binquet and Lee (1975) suggested that the average rate of corrosion of galvanized steel strips varies between 0.025 and 0.050 mm>yr. So, in the actual design of reinforcement, allowance must be made for the rate of corrosion. Thus, tc 5 tdesign 1 r (life span of structure) where tc 5 actual thickness of reinforcing strips to be used in construction tdesign 5 thickness of strips determined from design calculations r 5 rate of corrosion Further research needs to be done on corrosion-resistant materials such as fiberglass before they can be used as reinforcing strips. Nonbiodegradable Fabrics Nonbiodegradable fabrics are generally referred to as geotextiles. Since 1970, the use of geotextiles in construction has increased greatly around the world. The fabrics are usually made from petroleum products—polyester, polyethylene, and polypropylene. They may also be made from fiberglass. Geotextiles are not prepared from natural fabrics, because they decay too quickly. Geotextiles may be woven, knitted, or nonwoven. Woven geotextiles are made of two sets of parallel filaments or strands of yarn systematically interlaced to form a planar structure. Knitted geotextiles are formed by interlocking a series of loops of one or more filaments or strands of yarn to form a planar structure. Nonwoven geotextiles are formed from filaments or short fibers arranged in an oriented or random pattern in a planar structure. These filaments or short fibers are arranged into a loose web in the beginning and then are bonded by one or a combination of the following processes: 1. Chemical bonding—by glue, rubber, latex, a cellulose derivative, or the like 2. Thermal bonding—by heat for partial melting of filaments 3. Mechanical bonding—by needle punching Needle-punched nonwoven geotextiles are thick and have high in-plane permeability. Geotextiles have four primary uses in foundation engineering: 1. Drainage: The fabrics can rapidly channel water from soil to various outlets, thereby providing a higher soil shear strength and hence stability. 2. Filtration: When placed between two soil layers, one coarse grained and the other fine grained, the fabric allows free seepage of water from one layer to the other. However, it protects the fine-grained soil from being washed into the coarse-grained soil. 3. Separation: Geotextiles help keep various soil layers separate after construction and during the projected service period of the structure. For example, in the construction of highways, a clayey subgrade can be kept separate from a granular base course. 4. Reinforcement: The tensile strength of geofabrics increases the load-bearing capacity of the soil. 8.12 Considerations in Soil Reinforcement 407 Geogrids Geogrids are high-modulus polymer materials, such as polypropylene and polyethylene, and are prepared by tensile drawing. Netlon, Ltd., of the United Kingdom was the first producer of geogrids. In 1982, the Tensar Corporation, presently Tensar International Corporation, introduced geogrids into the United States. Commercially available geogrids may be categorized by manufacturing process, principally: extruded, woven, and welded. Extruded geogrids are formed using a thick sheet of polyethylene or polypropylene that is punched and drawn to create apertures and to enhance engineering properties of the resulting ribs and nodes. Woven geogrids are manufactured by grouping polymeric—usually polyester and polypropylene—and weaving them into a mesh pattern that is then coated with a polymeric lacquer. Welded geogrids are manufactured by fusing junctions of polymeric strips. Extruded geogrids have shown good performance when compared to other types for pavement reinforcement applications. Geogrids generally are of two types: (a) uniaxial and (b) biaxial. Figures 8.22a and b shows these two types of geogrids, which are produced by Tensar International Corporation. Uniaxial TENSAR grids are manufactured by stretching a punched sheet of extruded high-density polyethylene in one direction under carefully controlled conditions. The process aligns the polymer’s long-chain molecules in the direction of draw and results in a product with high one-directional tensile strength and a high modulus. Biaxial TENSAR grids are manufactured by stretching the punched sheet of polypropylene in two orthogonal directions. This process results in a product with high tensile strength and a high modulus in two perpendicular directions. The resulting grid apertures are either square or rectangular. The commercial geogrids currently available for soil reinforcement have nominal rib thicknesses of about 0.5 to 1.5 mm (0.02 to 0.06 in.) and junctions of about 2.5 to 5 mm (0.1 to 0.2 in.). The grids used for soil reinforcement usually have openings or apertures that are rectangular or elliptical. The dimensions of the apertures vary from about 25 to 150 mm (1 to 6 in.). Geogrids are manufactured so that the open areas of the grids are greater than 50% of the total area. They develop reinforcing strength at low strain levels, such as 2% (Carroll, 1988). Table 8.4 gives some properties of the TENSAR biaxial geogrids that are currently available commercially. 60 Roll Length (Longitudinal) Roll Length (Longitudinal) (b) Roll Width (Transverse) Roll Width (Transverse) (c) (a) Figure 8.22 Geogrid: (a) uniaxial; (b) biaxial; (c) with triangular apertures (Courtesy of Tensar International Corporation) 408 Chapter 8: Retaining Walls Table 8.4 Properties of TENSAR Biaxial Geogrids Geogrid Property Aperture size Machine direction Cross-machine direction Open area Junction Thickness Tensile modulus Machine direction Cross-machine direction Material Polypropylene Carbon black BX1000 BX1100 BX1200 25 mm (nominal) 33 mm (nominal) 70% (minimum) 25 mm (nominal) 33 mm (nominal) 74% (nominal) 25 mm (nominal) 33 mm (nominal) 77% (nominal) 2.3 mm (nominal) 2.8 mm (nominal) 4.1 mm (nominal) 182 kN>m (minimum) 182 kN>m (minimum) 204 kN>m (minimum) 292 kN>m (minimum) 270 kN>m (minimum) 438 kN>m (minimum) 97% (minimum) 2% (minimum) 99% (nominal) 1% (nominal) 99% (nominal) 1% (nominal) The major function of geogrids is reinforcement. They are relatively stiff. The apertures are large enough to allow interlocking with surrounding soil or rock (Figure 8.23) to perform the function of reinforcement or segregation (or both). Sarsby (1985) investigated the influence of aperture size on the size of soil particles for maximum frictional efficiency (or efficiency against pullout). According to this study, the highest efficiency occurs when BGG ⬎ 3.5D50 Figure 8.23 Geogrid apertures allowing interlocking with surrounding soil (8.30) 8.13 General Design Considerations 409 where BGG ⫽ minimum width of the geogrid aperture D50 ⫽ the particle size through which 50% of the backfill soil passes (i.e., the average particle size) More recently, geogrids with triangular apertures (Figure 8.22c) have been introduced for construction purposes. TENSAR geogrids with triangular apertures are manufactured from a punched polypropylene sheet, which is then oriented in three substantially equilateral directions so that the resulting ribs shall have a high degree of molecular orientation. Table 8.5 gives some properties of TENSAR geogrids with triangular apertures. Table 8.5 Properties of TENSAR Geogrids with Triangular Apertures Geogrid Property Longitudinal Diagonal Transverse TX 160 Rib pitch, (mm) Mid-rib depth, (mm) Mid-rib width, (mm) Nodal thickness, (mm) Radial stiffness at low strain, (kN/m @ 0.5% strain) Rib pitch, (mm) Mid-rib depth, (mm) Mid-rib width, (mm) Nodal thickness, (mm) Radial stiffness at low strain, (kN/m @ 0.5% strain) 40 — — 40 1.8 1.1 — 1.5 1.3 TX 170 8.13 General 3.1 430 40 — — 40 2.3 1.2 — 1.8 1.3 4.1 475 General Design Considerations The general design procedure of any mechanically stabilized retaining wall can be divided into two parts: 1. Satisfying internal stability requirements 2. Checking the external stability of the wall The internal stability checks involve determining tension and pullout resistance in the reinforcing elements and ascertaining the integrity of facing elements. The external stability checks include checks for overturning, sliding, and bearing capacity failure (Figure 8.24). The sections that follow will discuss the retaining-wall design procedures for use with metallic strips, geotextiles, and geogrids. 410 Chapter 8: Retaining Walls (a) Sliding (b) Overturning (c) Bearing capacity (d) Deep-seated stability Figure 8.24 External stability checks (After Transportation Research Board, 1995) (From Transportation Research Circular 444: Mechanically Stabilized Earth Walls. Transportation Research Board, National Research Council, Washington, D.C., 1995, Figure 3, p. 7. Reproduced with permission of the Transportation Research Board.) 8.14 Retaining Walls with Metallic Strip Reinforcement Reinforced-earth walls are flexible walls. Their main components are 1. Backfill, which is granular soil 2. Reinforcing strips, which are thin, wide strips placed at regular intervals, and 3. A cover or skin, on the front face of the wall Figure 8.25 is a diagram of a reinforced-earth retaining wall. Note that, at any depth, the reinforcing strips or ties are placed with a horizontal spacing of SH center to center; the vertical spacing of the strips or ties is SV center to center. The skin can be constructed with sections of relatively flexible thin material. Lee et al. (1973) showed that, with a conservative design, a 5 mm-thick galvanized steel skin would be enough to hold a wall about 14 to 15 m high. In most cases, precast concrete slabs can also be used as skin. The slabs are grooved to fit into each other so that soil cannot flow out between the joints. When metal skins are used, they are bolted together, and reinforcing strips are placed between the skins. Figures 8.26 and 8.27 show a reinforced-earth retaining wall under construction; its skin (facing) is a precast concrete slab. Figure 8.28 shows a metallic reinforcement tie attached to the concrete slab. The simplest and most common method for the design of ties is the Rankine method. We discuss this procedure next. 8.14 Retaining Walls with Metallic Strip Reinforcement 411 Tie Skin SH SV Figure 8.25 Reinforced-earth retaining wall Figure 8.26 Reinforced-earth retaining wall (with metallic strip) under construction (Courtesy of Braja M. Das, Henderson, NV) 412 Chapter 8: Retaining Walls Figure 8.27 Another view of the retaining wall shown in Figure 8.26 (Courtesy of Braja M. Das, Henderson, NV) Figure 8.28 Metallic strip attachment to the precast concrete slab used as the skin (Courtesy of Braja M. Das, Henderson, NV) 8.14 Retaining Walls with Metallic Strip Reinforcement 413 Calculation of Active Horizontal and Vertical Pressure Figure 8.29 shows a retaining wall with a granular backfill having a unit weight of g1 and a friction angle of f1r . Below the base of the retaining wall, the in situ soil has been excavated and recompacted, with granular soil used as backfill. Below the backfill, the in situ soil has a unit weight of g2 , friction angle of f2r , and cohesion of c2r . A surcharge having an intensity of q per unit area lies atop the retaining wall, which has reinforcement ties at depths z 5 0, SV , 2SV , c, NSV . The height of the wall is NSV 5 H. According to the Rankine active pressure theory (Section 7.3) sar 5 sor Ka 2 2cr"Ka where sar 5 Rankine active pressure at any depth z. For dry granular soils with no surcharge at the top, cr 5 0, sor 5 g1z, and Ka 5 tan2 (45 2 f1r >2). Thus, sa(1) r 5 g1zKa When a surcharge is added at the top, as shown in Figure 8.29, b a q/unit area C A 45 1/2 SV Sand ␥1 1 SV z lr (a) le SV H SV SV SV z NSV B In situ soil ␥2; 2; c2 (b) a(1) a(2) a Ka␥1z Figure 8.29 Analysis of a reinforced-earth retaining wall (8.31) 414 Chapter 8: Retaining Walls sor 5 so(1) r c 5 g1z Due to soil only 1 so(2) r c Due to the surcharge (8.32) The magnitude of so(2) r can be calculated by using the 2:1 method of stress distribution described in Eq. (5.14) and Figure 5.5. The 2:1 method of stress distribution is shown in Figure 8.30a. According to Laba and Kennedy (1986), so(2) r 5 qar ar 1 z (for z # 2br) (8.33) and so(2) r 5 qar z ar 1 1 br 2 (for z . 2br) (8.34) Also, when a surcharge is added at the top, the lateral pressure at any depth is r sar 5 sa(1) c 5 Kag1z Due to soil only b 1 sa(2) r c Due to the surcharge (8.35) b a a q/unit area q/unit area  z z H 2 2 1 1 Sand ␥1; 1 H o(2) Reinforcement strip (a) ␣ a(2) Sand ␥1; 1 Reinforcement strip (b) Figure 8.30 (a) Notation for the relationship of so(2) r in Eqs. (8.33) and (8.34); (b) notation for the relationship of sa(2) r in Eqs. (8.36) and (8.37) 415 8.14 Retaining Walls with Metallic Strip Reinforcement r may be expressed (see Figure 8.30b) as According to Laba and Kennedy (1986), sa(2) 2q (b 2 sin b cos 2a) R p sa(2) r 5 MB c (in radians) (8.36) where M 5 1.4 2 0.4br $1 0.14H (8.37) The net active (lateral) pressure distribution on the retaining wall calculated by using Eqs. (8.35), (8.36), and (8.37) is shown in Figure 8.29b. Tie Force The tie force per unit length of the wall developed at any depth z (see Figure 8.29) is T 5 active earth pressure at depth z 3 area of the wall to be supported by the tie 5 (sar ) (SVSH ) (8.38) Factor of Safety against Tie Failure The reinforcement ties at each level, and thus the walls, could fail by either (a) tie breaking or (b) tie pullout. The factor of safety against tie breaking may be determined as FS(B) 5 5 yield or breaking strength of each tie maximum force in any tie wtfy (8.39) sar SVSH where w 5 width of each tie t 5 thickness of each tie fy 5 yield or breaking strength of the tie material A factor of safety of about 2.5 to 3 is generally recommended for ties at all levels. Reinforcing ties at any depth z will fail by pullout if the frictional resistance developed along the surfaces of the ties is less than the force to which the ties are being subjected. The effective length of the ties along which frictional resistance is developed 416 Chapter 8: Retaining Walls may be conservatively taken as the length that extends beyond the limits of the Rankine active failure zone, which is the zone ABC in Figure 8.29. Line BC makes an angle of 45 1 f1r >2 with the horizontal. Now, the maximum friction force that can be realized for a tie at depth z is FR 5 2lewsor tan fmr (8.40) where le 5 effective length sor 5 effective vertical pressure at a depth z fmr 5 soil–tie friction angle Thus, the factor of safety against tie pullout at any depth z is FS(P) 5 FR T (8.41) Substituting Eqs. (8.38) and (8.40) into Eq. (8.41) yields FS(P) 5 2lewsor tan fmr sar SVSH (8.42) Total Length of Tie The total length of ties at any depth is L 5 lr 1 le (8.43) where lr 5 length within the Rankine failure zone le 5 effective length For a given FS(P) from Eq. (8.42), le 5 FS(P)sar SVSH 2wsor tan fmr (8.44) Again, at any depth z, lr 5 (H 2 z) f1r tan ¢45 1 ≤ 2 (8.45) So, combining Eqs. (8.43), (8.44), and (8.45) gives L5 (H 2 z) f1r tan ¢ 45 1 ≤ 2 1 FS(P)sar SVSH 2wsor tan fmr (8.46) 417 8.15 Step-by-Step-Design Procedure Using Metallic Strip Reinforcement 8.15 Step-by-Step-Design Procedure Using Metallic Strip Reinforcement Following is a step-by-step procedure for the design of reinforced-earth retaining walls. General Step 1. Determine the height of the wall, H, and the properties of the granular backfill material, such as the unit weight (g1 ) and the angle of friction (f1r ). Step 2. Obtain the soil–tie friction angle, fmr , and the required value of FS(B) and FS(P) . Internal Stability Step 3. Assume values for horizontal and vertical tie spacing. Also, assume the width of reinforcing strip, w, to be used. Step 4. Calculate sar from Eqs. (8.35), (8.36), and (8.37). Step 5. Calculate the tie forces at various levels from Eq. (8.38). Step 6. For the known values of FS(B) , calculate the thickness of ties, t, required to resist the tie breakout: T 5 sar SVSH 5 wtfy FS(B) or t5 (sar SVSH ) 3FS(B) 4 wfy (8.47) The convention is to keep the magnitude of t the same at all levels, so sar in Eq. (8.47) should equal sa(max) r . Step 7. For the known values of fmr and FS(P) , determine the length L of the ties at various levels from Eq. (8.46). Step 8. The magnitudes of SV , SH , t, w, and L may be changed to obtain the most economical design. External Stability Step 9. Check for overturning, using Figure 8.31 as a guide. Taking the moment about B yields the overturning moment for the unit length of the wall: Mo 5 Pazr (8.48) Here, H Pa 5 active force 5 3 sar dz 0 The resisting moment per unit length of the wall is 418 Chapter 8: Retaining Walls b a q/unit area A I L L1 x1 z W1 Sand ␥1; 1 F E G Pa L L2 H x2 z W2 B Sand ␥ ; 1 1 D Figure 8.31 Stability check for the retaining wall In situ soil ␥2; 2; c2 ar MR 5 W1x1 1 W2x2 1 c 1 qar ¢ br 1 ≤ 2 (8.49) where W1 5 (area AFEGI) (1) (g1 ) W2 5 (area FBDE) (1) (g1 ) .( So, FS(overturning) 5 MR Mo 5 ar W1x1 1 W2x2 1 c1 qar ¢ br 1 ≤ 2 (8.50) H ¢ 3 sar dz ≤ zr 0 Step 10. The check for sliding can be done by using Eq. (8.11), or FS(sliding) 5 where k < 23 . (W1 1 W2 1 c 1 qar) 3 tan (kf1r ) 4 Pa (8.51) 8.15 Step-by-Step-Design Procedure Using Metallic Strip Reinforcement 419 Step 11. Check for ultimate bearing capacity failure, which can be given as qu 5 c2r Nc 1 12g2L2Ng (8.52) The bearing capacity factors Nc and Ng correspond to the soil friction angle f2r . (See Table 3.3.) From Eq. 8.32, the vertical stress at z 5 H is so(H) r 5 g1H 1 so(2) r (8.53) So the factor of safety against bearing capacity failure is FS(bearing capacity) 5 qult so(H) r (8.54) Generally, minimum values of FS(overturning) 5 3, FS(sliding) 5 3, and FS(bearing capacity failure) 5 3 to 5 are recommended. Example 8.5 A 10 m high retaining wall with galvanized steel-strip reinforcement in a granular backfill has to be constructed. Referring to Figure 8.29, given: Granular backfill: f1r 5 36° g1 5 16.5 kN>m3 Foundation soil: f2r 5 28° g2 5 17.3 kN>m3 c2r 5 50 kN>m2 Galvanized steel reinforcement: Width of strip, w 5 75 mm SV 5 0.6 m center-to-center SH 5 1 m center-to-center fy 5 240,00 kN>m2 fmr 5 20° Required FS(B) 5 3 Required FS(P) 5 3 Check for the external and internal stability. Assume the corrosion rate of the galvanized steel to be 0.025 mm> year and the life span of the structure to be 50 years. Solution Internal Stability Check r SVSH Tie thickness: Maximum tie force, Tmax 5 sa(max) 420 Chapter 8: Retaining Walls sa(max) 5 g1HKa 5 g1H tan2 a45 2 f1r b 2 so Tmax 5 g1H tan2 a45 2 f1r bS S 2 V H From Eq. (8.47), for tie break, t5 (sar SVSH ) 3FS(B) 4 wfy 5 f1r bS S d FS(B) 2 V H wfy cg1H tan2 a45 2 or t5 c (16.5) (10) tan2 a45 2 36 b (0.6) (1) d (3) 2 5 0.00428m 5 4.28 mm (0.075 m) (240,000 kN>m2 ) If the rate of corrosion is 0.025 mm> yr and the life span of the structure is 50 yr, then the actual thickness, t, of the ties will be t 5 4.28 1 (0.025) (50) 5 5.53 mm So a tie thickness of 6 mm would be enough. Tie length: Refer to Eq. (8.46). For this case, sar 5 g1zKa and sor 5 g1z, so FS(P)g1zKaSVSH (H 2 z) 1 f1r 2wg1z tanfmr tana45 1 b 2 L5 Now the following table can be prepared. (Note: FS(P) 5 3, H 5 10 m, w 5 75 mm, and fmr 5 20°.) z (m) Tie length L (m) [Eq. (8.46)] 2 4 6 8 10 12.65 11.63 10.61 9.59 8.57 So use a tie length of L 5 13 m. External Stability Check Check for overturning: Refer to Figure 8.32. For this case, using Eq. (8.50) FS(overturning) 5 W1x1 H c 3 sar dz d zr 0 8.15 Step-by-Step-Design Procedure Using Metallic Strip Reinforcement 421 ␥1 16.5 kN/m3 1 36° 6.5 m 10 m W1 L 13 m ␥2 17.3 kN/m3 2 28° c2 50 kN/m2 Figure 8.32 Retaining wall with galvanized steel-strip reinforcement in the backfill W1 5 g1HL 5 (16.5) (10) (13) 5 2145 kN>m x1 5 6.5 m H Pa 5 3 sar dz 5 12g1KaH 2 5 ( 12 ) (16.5) (0.26) (10) 2 5 214.5 kN>m 0 zr 5 FS(overturning) 5 10 5 3.33 m 3 (2145) (6.5) 5 19.52 + 3—OK (214.5) (3.33) Check for sliding: From Eq. (8.51) FS(sliding) W1 tan(kf1r ) 5 5 Pa 2 2145 tan c a b (36) d 3 5 4.45 + 3—OK 214.5 Check for bearing capacity: For f2r 5 28°, Nc 5 25.8, Ng 5 16.78 (Table 3.3). From Eq. (8.52), qult 5 c2r Nc 1 12g2L Ng qult 5 (50) (25.8) 1 ( 12 ) (17.3) (13) (16.72) 5 3170.16 kN>m2 From Eq. (8.53), so(H) r 5 g1H 5 (16.5) (10) 5 165 kN>m2 FS(bearing capacity) 5 qult 3170.16 5 5 19.2 + 5—OK so(H) r 165 ■ 422 Chapter 8: Retaining Walls 8.16 Retaining Walls with Geotextile Reinforcement Figure 8.33 shows a retaining wall in which layers of geotextile have been used as reinforcement. As in Figure 8.31, the backfill is a granular soil. In this type of retaining wall, the facing of the wall is formed by lapping the sheets as shown with a lap length of ll . When construction is finished, the exposed face of the wall must be covered; otherwise, the geotextile will deteriorate from exposure to ultraviolet light. Bitumen emulsion or Gunite is sprayed on the wall face. A wire mesh anchored to the geotextile facing may be necessary to keep the coating on. Figure 8.34 shows the construction of a geotextile-reinforced retaining wall. Figure 8.35 shows a completed geosynthetic-reinforced soil wall. The wall is in DeBeque Canyon, Colorado. Note the versatility of the facing type. In this case, single-tier concrete block facing is integrated with a three-tier facing via rock facing. SV Geotextile SV z lr H 45 1/2 In situ soil ␥2; 2; c2 ll SV Geotextile le SV Sand,␥1; 1 Geotextile SV Geotextile Geotextile Figure 8.33 Retaining wall with geotextile reinforcement Figure 8.34 Construction of a geotextile-reinforced retaining wall (Courtesy of Jonathan T. H. Wu, University of Colorado at Denver, Denver, Colorado) 423 8.16 Retaining Walls with Geotextile Reinforcement Figure 8.35 A completed geotextile-reinforced retaining wall in DeBeque Canyon, Colorado (Courtesy of Jonathan T. H. Wu, University of Colorado at Denver, Denver, Colorado) The design of this type of retaining wall is similar to that presented in Section 8.15. Following is a step-by-step procedure for design based on the recommendations of Bell et al. (1975) and Koerner (2005): Internal Stability Step 1. Determine the active pressure distribution on the wall from the formula sar 5 Kasor 5 Kag1z (8.55) where Ka 5 Rankine active pressure coefficient 5 tan2 (45 2 f1r >2) g1 5 unit weight of the granular backfill f1r 5 friction angle of the granular backfill Step 2. Select a geotextile fabric with an allowable tensile strength, Tall (lb>ft or kN>m ). The allowable tensile strength for retaining wall construction may be expressed as (Koerner, 2005) Tall 5 Tult RFid 3 RFcr 3 RFcbd where Tult ⫽ ultimate tensile strength RFid ⫽ reduction factor for installation damage RFcr ⫽ reduction factor for creep RFcbd ⫽ reduction factor for chemical and biological degradation (8.56) 424 Chapter 8: Retaining Walls The recommended values of the reduction factor are as follows (Koerner, 2005) RFid RFcr RFcbd 1.1–2.0 2–4 1–1.5 Step 3. Determine the vertical spacing of the layers at any depth z from the formula SV 5 Tall Tall 5 sar FS(B) (g1zKa ) 3FS(B) 4 (8.57) Note that Eq. (8.57) is similar to Eq. (8.39). The magnitude of FS(B) is generally 1.3 to 1.5. Step 4. Determine the length of each layer of geotextile from the formula L 5 lr 1 le (8.58) where lr 5 H2z (8.59) f1r tan ¢45 1 ≤ 2 and le 5 SVsar 3FS(P) 4 (8.60) 2sor tan fFr in which sar 5 g1zKa sor 5 g1z FS(P) 5 1.3 to 1.5 fFr 5 friction angle at geotextile–soil interface < 23f1r Note that Eqs. (8.58), (8.59), and (8.60) are similar to Eqs. (8.43), (8.45), and (8.44), respectively. Based on the published results, the assumption of fFr >f1r < 23 is reasonable and appears to be conservative. Martin et al. (1984) presented the following laboratory test results for fFr >f1r between various types of geotextiles and sand. Type Woven—monofilament> concrete sand Woven—silt film> concrete sand Woven—silt film> rounded sand Woven—silt film> silty sand Nonwoven—melt-bonded> concrete sand Nonwoven—needle-punched> concrete sand Nonwoven—needle-punched> rounded sand Nonwoven—needle-punched> silty sand fF9 , f19 0.87 0.8 0.86 0.92 0.87 1.0 0.93 0.91 8.16 Retaining Walls with Geotextile Reinforcement Step 5. Determine the lap length, ll , from SVsar FS(P) ll 5 4sor tan fFr 425 (8.61) The minimum lap length should be 1 m. External Stability Step 6. Check the factors of safety against overturning, sliding, and bearing capacity failure as described in Section 8.15 (Steps 9, 10, and 11). Example 8.6 A geotextile-reinforced retaining wall 5 m high is shown in Figure 8.36. For the granular backfill, g1 5 15.7 kN>m3 and f1r 5 36°. For the geotextile, Tult 5 52.5 kN>m. For the design of the wall, determine SV , L, and ll . Use RFid 5 1.2, RFcr 5 2.5, and RFcbd 5 1.25. Solution We have Ka 5 tan2 ¢ 45 2 f1r ≤ 5 0.26 2 Determination of SV To find SV , we make a few trials. From Eq. (8.57), Tall SV 5 (g1zKa ) 3FS(B) 4 2.5 m SV = 0.5 m 5m ␥1 = 15.7 kN/m3 1 = 36° ll = l m ␥2 = 18 kN/m3 2 = 22° c2 = 28 kN/m2 Figure 8.36 Geotextile-reinforced retaining wall 426 Chapter 8: Retaining Walls From Eq. (8.56), Tall 5 Tuef 52.5 5 5 14 kN>m RFid 3 RFcr 3 RFcbd 1.2 3 2.5 3 1.25 With FS(B) 5 1.5 at z 5 2 m, SV 5 14 5 1.14 m (15.7) (2) (0.26) (1.5) SV 5 14 5 0.57 m (15.7) (4) (0.26) (1.5) SV 5 14 5 0.46 m (15.7) (5) (0.26) (1.5) At z 5 4 m, At z 5 5 m, So, use SV 5 0.5 m for z 5 0 to z 5 5 m (See Figure 8.36.) Determination of L From Eqs. (8.58), (8.59), and (8.60), (H 2 z) L5 tan ¢ 45 1 f1r ≤ 2 1 SVKa 3FS(P) 4 2 tan fFr For FS(P) 5 1.5, tan fFr 5 tan 3( 23 ) (36)4 5 0.445, and it follows that L 5 (0.51) (H 2 z) 1 0.438SV H ⫽ 5 m, SV ⫽ 0.5 m At z ⫽ 0.5 m: L ⫽ (0.51)(5 ⫺ 0.5) ⫹ (0.438)(0.5) ⫽ 2.514 m At z ⫽ 2.5 m: L ⫽ (0.51)(5 ⫺ 2.5) ⫹ (0.438)(0.5) ⫽ 1.494 m So, use L ⴝ 2.5 m throughout. Determination of ll From Eq. (8.61), ll 5 SVsar 3FS(P) 4 4sor tan fFr sar 5 g1zKa , FS(P) 5 1.5; with sor 5 g1z, fFr 5 23f1r . So ll 5 SVKa 3FS(P) 4 So, use l l 5 1 m. 5 SV (0.26) (1.5) 5 0.219SV 4 tan 3( 23 ) (36) 4 ll 5 0.219SV 5 (0.219) (0.5) 5 0.11 m # 1 m 4 tan fFr ■ 8.16 Retaining Walls with Geotextile Reinforcement 427 Example 8.7 Consider the results of the internal stability check given in Example 8.6. For the geotextile-reinforced retaining wall, calculate the factor of safety against overturning, sliding, and bearing capacity failure. Solution Refer to Figure 8.37. Factor of Safety Against Overturning From Eq. (8.50), FS (overturning) 5 W1x1 (Pa ) a H b 3 W1 ⫽ (5)(2.5)(15.7) ⫽ 196.25 kN/m x1 5 2.5 5 1.25 m 2 1 1 Pa 5 gH 2Ka 5 a b (15.7) (5) 2 (0.26) 5 51.03 kN>m 2 2 Hence, FS (overturning) 5 (196.25) (1.25) 5 2.88 , 3 51.03(5>3) (increase length of geotextile layers to 3 m) 2.5 m SV = 0.5 m x1 5m W1 ␥1 = 15.7 kN/m3 1 = 36° ll = 1 m ␥2 = 18 kN/m3 2 = 22° c2 = 28 kN/m2 Figure 8.37 Stability check 428 Chapter 8: Retaining Walls Factor of Safety Against Sliding From Eq. (8.51), FS (sliding) 2 2 W1tana f1rb (196.25) ctana 3 36b d 3 3 5 5 5 1.71 + 1.5 2 O.K. Pa 51.03 Factor of Safety Against Bearing Capacity Failure 1 From Eq. (8.52), qu 5 c2r Nc 1 g2 L2 Ng 2 Given: ␥2 ⫽ 18 kN/m3, L2 ⫽ 2.5 m, c2⬘ ⫽ 28 kN/m2, and 2⬘ ⫽ 22°. From Table 3.3, Nc ⫽ 16.88, and N␥ ⫽ 7.13. 1 qu 5 (28) (16.88) 1 a b (18) (2.5) (7.13) < 633 kN>m2 2 From Eq. (8.54), FS(bearing capacity) 5 8.17 qu so9(H) 5 633 633 5 5 8.06 + 3 2 O.K. g1H (15.7) (5) ■ Retaining Walls with Geogrid Reinforcement—General Geogrids can also be used as reinforcement in granular backfill for the construction of retaining walls. Figure 8.38 shows typical schematic diagrams of retaining walls with geogrid reinforcement. Figure 8.39 shows some photographs of geogrid-reinforced retaining walls in the field. Relatively few field measurements are available for lateral earth pressure on retaining walls constructed with geogrid reinforcement. Figure 8.40 shows a comparison of measured and design lateral pressures (Berg et al., 1986) for two retaining walls constructed with precast panel facing. The figure indicates that the measured earth pressures were substantially smaller than those calculated for the Rankine active case. 8.18 Design Procedure for Geogrid-Reinforced Retaining Wall Figure 8.41 shows a schematic diagram of a concrete panel-faced wall with a granular backfill reinforced with layers of geogrid. The design process of the wall is essentially similar to that with geotextile reinforcement of the backfill given in Section 8.16. The following is a brief step-by-step procedure. Internal Stability Step 1. Determine the active pressure at any depth z as [similar to Eq. (8.55)]: a⬘ ⫽ Ka␥1z (8.62) where Ka ⫽ Rankine active pressure coefficient ⫽ tan2 a45 2 f91 b 2 8.18 Design Procedure fore Geogrid-Reinforced Retaining Wall 429 Geogrids – biaxial Geogrids – uniaxial (a) Gabion facing Geogrids (b) Precast concrete panel Pinned connection Geogrids Leveling pad (c) Figure 8.38 Typical schematic diagrams of retaining walls with geogrid reinforcement: (a) geogrid wraparound wall; (b) wall with gabion facing; (c) concrete panel-faced wall (After The Tensar Corporation, 1986) 430 Chapter 8: Retaining Walls (b) (a) (c) 0 Figure 8.39 (a) HDPE geogridreinforced wall with precast concrete panel facing under construction; (b) Mechanical splice between two pieces of geogrid in the working direction; (c) Segmented concrete-block faced wall reinforced with uniaxial geogrid (Courtesy of Tensar International Corporation, Atlanta, Georgia) Lateral pressure, a (kN/m2) 10 20 30 40 0 Wall at Tuscon, Arizona, H 4.6 m 1 2 3 Wall at Lithonia, Georgia, H 6 m Measured pressure Rankine active pressure 4 5 Height of fill above load cell (m) Figure 8.40 Comparison of theoretical and measured lateral pressures in geogrid reinforced retaining walls (Based on Berg et al., 1986) 8.18 Design Procedure fore Geogrid-Reinforced Retaining Wall 431 W1 z Granular backfill SV L1 ␥1 1 H W2 L2 Leveling pad Foundation soil ␥2, 2, c2 Figure 8.41 Design of geogrid-reinforced retaining wall Step 2. Select a geogrid with allowable tensile strength, Tall [similar to Eq. (8.56)] (Koerner, 2005): Tult Tall 5 (8.63) RFid 3 RFcr 3 RFcbd where RFid ⫽ reduction factor for installation damage (1.1 to 1.4) RFcr ⫽ reduction factor for creep (2.0 to 3.0) RFcbd ⫽ reduction factor for chemical and biological degradation (1.1 to 1.5). Step 3. Obtain the vertical spacing of the geogrid layers, SV, as SV 5 TallCr sra FS(B) (8.64) where Cr ⫽ coverage ratio for geogrid. The coverage ratio is the fractional plan area at any particular elevation that is actually occupied by geogrid. For example, if there is a 0.3 m (1 ft) wide space between each 1.2 m (4 ft) wide piece of geogrid, the coverage ratio is Cr 5 1.2 m 5 0.8 1.2 m 1 0.3 m Step 4. Calculate the length of each layer of geogrid at a depth z as [Eq. (8.58)] L ⫽ lr ⫹ le lr 5 H2z tan2 a45 2 fr1 b 2 (8.65) 432 Chapter 8: Retaining Walls For determination of le [similar to Eq. (8.60)], FS(P) 5 resistance to pullout at a given normal effective stress pullout force 5 (2) (le ) (Cis0r tan f1r) (Cr ) SVsar 5 (2) (le ) (Ci tan f1r) (Cr ) SVKa (8.66) where Ci ⫽ interaction coefficient or le 5 SVKa FS(P) (8.67) 2CrCi tan fr1 Thus, at a given depth z, the total length, L, of the geogrid layer is L 5 lr 1 le 5 SVKa FS(P) H2z 1 2CrCi tan fr1 f1r tana45 1 b 2 (8.68) The interaction coefficient, Ci, can be determined experimentally in the laboratory. The following is an approximate range for Ci for various types of backfill. Gravel, sandy gravel Well graded sand, gravelly sand Fine sand, silty sand 0.75–0.8 0.7–0.75 0.55–0.6 External Stability Check the factors of safety against overturning, sliding, and bearing capacity failure as described in Section 8.15 (Steps 9, 10, and 11). Example 8.8 Consider a geogrid-reinforced retaining wall. Referring to Figure 8.41, given: H ⫽ 6 m, ␥1 ⫽ 16.5 kN/m3, 1⬘ ⫽ 35°, Tall ⫽ 45 kN/m, FS(B) ⫽ 1.5, FS(P) ⫽ 1.5, Cr ⫽ 0.8, and Ci ⫽ 0.75. For the design of the wall, determine SV and L. Solution Ka 5 tan2 a45 2 fr1 35 b 5 tan2 a45 2 b 5 0.27 2 2 Determination of SV From Eq. (8.64), Sv 5 TallCr s9a FS(B) 5 TallCr (45) (0.8) 5.39 5 5 z gzKa FS(B) (16.5) (z) (0.27) (1.5) Problems 433 5.39 5 2.7 m 2 5.39 At z ⴝ 4 m: Sv 5 5 1.35 m 4 5.39 At z ⴝ 5 m: Sv 5 5 1.08 m 5 Use SV ⬇ 1 m At z ⴝ 2 m: Sv 5 Determination of L From Eq. (8.68), L5 SVKa FS(P) (1 m) (0.27) (1.5) 62z H2z 1 5 1 2CrCi tanf1r (2) (0.8) (0.75) (tan 35°) f1r 35 tana45 1 b tana45 1 b 2 2 At z ⴝ 1 m: L ⴝ 0.52(6 ⫺ 1) ⫹ 0.482 ⴝ 3.08 m ⬇ 3.1 m At z ⴝ 3 m: L ⴝ 0.52(6 ⫺ 3) ⫹ 0.482 ⴝ 2.04 m ⬇ 2.1 m At z ⴝ 5 m: L ⴝ 0.52(6 ⫺ 5) ⫹ 0.482 ⴝ 1.0 m So, use L ⴝ 3 m for z ⴝ 0 to 6 m. ■ Problems In Problems 8.1 through 8.4, use gconcrete ⫽ 23.58 kN> m3. Also, in Eq. (8.11), use k1 ⫽ k2 ⫽ 2> 3 and Pp ⫽ 0. 8.1 For the cantilever retaining wall shown in Figure P8.1, let the following data be given: Wall dimensions: Soil properties: 8.2 H ⫽ 8 m, x1 ⫽ 0.4 m, x2 ⫽ 0.6 m, x3 ⫽ 1.5 m, x4 ⫽ 3.5 m, x5 ⫽ 0.96 m, D ⫽ 1.75 m, a ⫽ 10° g1 ⫽ 16.5 kN> m3, f1r ⫽ 32°, g2 ⫽ 17.6 kN> m3, f2r 5 28°, cr2 ⫽ 30 kN> m2 Calculate the factor of safety with respect to overturning, sliding, and bearing capacity. Repeat Problem 8.1 with the following: Wall dimensions: H ⫽ 6.5 m, x1 ⫽ 0.3 m, x2 ⫽ 0.6 m, x3 ⫽ 0.8 m, x4 ⫽ 2 m, x5 ⫽ 0.8 m, D ⫽ 1.5 m, a ⫽ 0° g1 ⫽ 18.08 kN/m3, f1r ⫽ 36°, g2 ⫽ 19.65 kN/m3, f2r ⫽ 15°,cr2 ⫽ 30 kN> m2 A gravity retaining wall is shown in Figure P8.3. Calculate the factor of safety with respect to overturning and sliding, given the following data: Soil properties: 8.3 Wall dimensions: Soil properties: H ⫽ 6 m, x1 ⫽ 0.6 m, x2 ⫽ 2 m, x3 ⫽ 2 m, x4 ⫽ 0.5 m, x5 ⫽ 0.75 m, x6 ⫽ 0.8 m, D ⫽ 1.5 m g1 ⫽ 16.5 kN> m3, f1r ⫽ 32°, g2 ⫽ 18 kN/m3, f2r ⫽ 22°, c2r ⫽ 40 kN> m2 Use the Rankine active earth pressure in your calculation. 434 Chapter 8: Retaining Walls ␣ x1 ␥1 c1 0 1 H D x5 x2 x3 x4 ␥2 2 c2 H ␥1 c1 0 1 x5 x6 D x4 x2 x1 x3 ␥2 2 c2 8.4 Figure P8.1 Figure P8.3 Repeat Problem 8.3 using Coulomb’s active earth pressure in your calculation and letting d r ⫽ 2> 3 f1r . 8.5 Refer to Figure P8.5 for the design of a gravity retaining wall for earthquake condition Given: kv ⫽ 0 and kh ⫽ 0.3. a. What should be the weight of the wall for a zero displacement condition? Use a factor of safety of 2. b. What should be the weight of the wall for an allowable displacement of 50.8 mm? References 7m Sand 8.6 8.7 8.8 435 Sand 1 30 ␥1 18 kN/m3 ␦1 15 2 36 ␥2 18.5 kN/m3 Figure P8.5 Given: Av ⫽ 0.15 and Aa ⫽ 0.25. Use a factor of safety of 2. In Figure 8.29a, use the following parameters: Wall: H ⫽ 8 m Soil: g1 ⫽ 17 kN> m3, f1r ⫽ 35° Reinforcement: SV ⫽ 1 m and SH ⫽ 1.5 m Surcharge: q ⫽ 70 kN> m2, a r ⫽ 1.5 m, and b r ⫽ 2 m Calculate the vertical stress sor [Eqs. (8.32), (8.33) and (8.34)] at z ⫽ 2 m, 4 m, 6 m and 8 m. For the data given in Problem 8.6, calculate the lateral pressure sar at z ⫽ 2 m, 4 m, 6 m and 8 m. Use Eqs. (8.35), (8.36) and (8.37). A reinforced earth retaining wall (Figure 8.29) is to be 10 m. high. Here, Backfill: unit weight, g1 ⫽ 16 kN> m3 and soil friction angle, f1r ⫽ 34° Reinforcement: vertical spacing, SV 5 1 m; horizontal spacing, SH 5 1.25 m; width of reinforcement 5 120 mm., fy 5 260 MN>m2; fm 5 25°; factor of safety against tie pullout 5 3; and factor of safety against tie breaking ⫽ 3 Determine: a. The required thickness of ties b. The required maximum length of ties 8.9 In Problem 8.8 assume that the ties at all depths are the length determined in Part b. For the in situ soil, f2r ⫽ 25°, g2 ⫽ 15.5 kN> m3, c2r ⫽ 30 kN> m2. Calculate the factor of safety against (a) overturning, (b) sliding, and (c) bearing capacity failure. 8.10 A retaining wall with geotextile reinforcement is 6-m high. For the granular backfill, g1 ⫽ 15.9 kN> m3 and f1r ⫽ 30°. For the geotextile, Tall ⫽ 16 kN> m. For the design of the wall, determine SV, L, and ll. Use FS(B) ⫽ FS(P) ⫽ 1.5. 8.11 With the SV, L, and ll determined in Problem 8.10, check the overall stability (i.e., factor of safety against overturning, sliding, and bearing capacity failure) of the wall. For the in situ soil, g2 ⫽ 16.8 kN> m3, f2r ⫽ 20°, and c2r ⫽ 55 kN> m2. References APPLIED TECHNOLOGY COUNCIL (1978). “Tentative Provisions for the Development of Seismic Regulations for Buildings,” Publication ATC 3-06, Palo Alto, California. BELL, J. R., STILLEY, A. N., and VANDRE, B. (1975). “Fabric Retaining Earth Walls,” Proceedings, Thirteenth Engineering Geology and Soils Engineering Symposium, Moscow, ID. 436 Chapter 8: Retaining Walls BENTLER, J. G., and LABUZ, J. F. (2006). “Performance of a Cantilever Retaining Wall,” Journal of Geotechnical and Geoenvironmental Engineering, American Society of Civil Engineers, Vol. 132, No. 8, pp. 1062–1070. BERG, R. R., BONAPARTE, R., ANDERSON, R. P., and CHOUERY, V. E. (1986). “Design Construction and Performance of Two Tensar Geogrid Reinforced Walls,” Proceedings, Third International Conference on Geotextiles, Vienna, pp. 401– 406. BINQUET, J., and LEE, K. L. (1975). “Bearing Capacity Analysis of Reinforced Earth Slabs,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 101, No. GT12, pp. 1257–1276. CARROLL, R., JR. (1988). “Specifying Geogrids,” Geotechnical Fabric Report, Industrial Fabric Association International, St. Paul, March/April. CASAGRANDE, L. (1973). “Comments on Conventional Design of Retaining Structure,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 99, No. SM2, pp. 181–198. DARBIN, M. (1970). “Reinforced Earth for Construction of Freeways” (in French), Revue Générale des Routes et Aerodromes, No. 457, September. DAS, B. M. (1983). Fundamentals of Soil Dynamics, Elsevier, New York. ELMAN, M. T., and TERRY, C. F. (1988). “Retaining Walls with Sloped Heel,” Journal of Geotechnical Engineering, American Society of Civil Engineers, Vol. 114, No. GT10, pp. 1194–1199. KOERNER, R. B. (2005). Design with Geosynthetics, 5th. ed., Prentice Hall, Englewood Cliffs, NJ. LABA, J. T., and KENNEDY, J. B. (1986). “Reinforced Earth Retaining Wall Analysis and Design,” Canadian Geotechnical Journal, Vol. 23, No. 3, pp. 317–326. LEE, K. L., ADAMS, B. D., and VAGNERON, J. J. (1973). “Reinforced Earth Retaining Walls,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 99, No. SM10, pp. 745 –763. MARTIN, J. P., KOERNER, R. M., and WHITTY, J. E. (1984). “Experimental Friction Evaluation of Slippage Between Geomembranes, Geotextiles, and Soils,” Proceedings, International Conference on Geomembranes, Denver, pp. 191–196. RICHARDS, R., and ELMS, D. G. (1979). “Seismic Behavior of Gravity Retaining Walls,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 105, No. GT4, pp. 449–464. SARSBY, R. W. (1985). “The Influence of Aperture Size/Particle Size on the Efficiency of Grid Reinforcement,” Proceedings, 2nd Canadian Symposium on Geotextiles and Geomembranes, Edmonton, pp. 7–12. SCHLOSSER, F., and LONG, N. (1974). “Recent Results in French Research on Reinforced Earth,” Journal of the Construction Division, American Society of Civil Engineers, Vol. 100, No. CO3, pp. 113 –237. SCHLOSSER, F., and VIDAL, H. (1969). “Reinforced Earth” (in French), Bulletin de Liaison des Laboratoires Routier, Ponts et Chassées, Paris, France, November, pp. 101–144. TENSAR CORPORATION (1986). Tensar Technical Note. No. TTN:RW1, August. TERZAGHI, K., and PECK, R. B. (1967). Soil Mechanics in Engineering Practice, Wiley, New York. TRANSPORTATION RESEARCH BOARD (1995). Transportation Research Circular No. 444, National Research Council, Washington, DC. VIDAL, H. (1966). “La terre Armée,” Annales de l’Institut Technique du Bâtiment et des Travaux Publiques, France, July–August, pp. 888–938. 9 9.1 Sheet Pile Walls Introduction Connected or semiconnected sheet piles are often used to build continuous walls for waterfront structures that range from small waterfront pleasure boat launching facilities to large dock facilities. (See Figure 9.1.) In contrast to the construction of other types of retaining wall, the building of sheet pile walls does not usually require dewatering of the site. Sheet piles are also used for some temporary structures, such as braced cuts. (See Chapter 10.) The principles of sheet-pile wall design are discussed in the current chapter. Several types of sheet pile are commonly used in construction: (a) wooden sheet piles, (b) precast concrete sheet piles, and (c) steel sheet piles. Aluminum sheet piles are also marketed. Wooden sheet piles are used only for temporary, light structures that are above the water table. The most common types are ordinary wooden planks and Wakefield piles. The wooden planks are about 50 mm 3 300 mm in cross section and are driven edge to edge (Figure 9.2a). Wakefield piles are made by nailing three planks together, with the middle plank offset by 50 to 75 mm (Figure 9.2b). Wooden planks can also be milled to form tongue-and-groove piles, as shown in Figure 9.2c. Figure 9.2d shows another type of wooden sheet pile that has precut grooves. Metal splines are driven into the grooves of the adjacent sheetings to hold them together after they are sunk into the ground. Water table Sheet pile Water table Land side Dredge line Figure 9.1 Example of waterfront sheet-pile wall 437 438 Chapter 9: Sheet Pile Walls Precast Concrete Sheet Pile Wooden Sheet Piles 150250 mm (a) Planks Concrete grout Section 500-800 mm (b) Wakefield piles Reinforcement (c) Tongue-and-groove piles Elevation (e) (d) Splined piles (not to scale) Figure 9.2 Various types of wooden and concrete sheet pile Precast concrete sheet piles are heavy and are designed with reinforcements to withstand the permanent stresses to which the structure will be subjected after construction and also to handle the stresses produced during construction. In cross section, these piles are about 500 to 800 mm wide and 150 to 250 mm thick. Figure 9.2e is a schematic diagram of the elevation and the cross section of a reinforced concrete sheet pile. Steel sheet piles in the United States are about 10 to 13 mm thick. European sections may be thinner and wider. Sheet-pile sections may be Z, deep arch, low arch, or straight web sections. The interlocks of the sheet-pile sections are shaped like a thumband-finger or ball-and-socket joint for watertight connections. Figure 9.3a is a schematic diagram of the thumb-and-finger type of interlocking for straight web sections. The balland-socket type of interlocking for Z section piles is shown in Figure 9.3b. Figure 9.4 shows a sheet pile wall. Table 9.1 lists the properties of the steel sheet pile sections produced by the Bethlehem Steel Corporation. The allowable design flexural stress for the steel sheet piles is as follows: Type of steel Allowable stress ASTM A-328 ASTM A-572 ASTM A-690 170 MN>m2 210 MN>m2 210 MN>m2 Steel sheet piles are convenient to use because of their resistance to the high driving stress that is developed when they are being driven into hard soils. Steel sheet piles are also lightweight and reusable. 9.1 Introduction 439 (a) Figure 9.3 (a) Thumb-and-finger type sheet pile connection; (b) ball-and-socket type sheet-pile connection (b) (c) Figure 9.4 A steel sheet pile wall (Courtesy of N. Sivakugan, James Cook University, Australia ) Table 9.1 Properties of Some Sheet-Pile Sections Produced by Bethlehem Steel Corporation Section modulus Moment of inertia m ,m of wall m4 , m of wall 3 Section designation Sketch of section 326.4 3 1025 PZ-40 670.5 3 1026 12.7 mm 409 mm 15.2 mm Driving distance 500 mm (Continued) 440 Chapter 9: Sheet Pile Walls Table 9.1 (Continued) Section designation Sketch of section PZ-35 Section modulus Moment of inertia m3 , m of wall m4 , m of wall 260.5 3 1025 493.4 3 1026 162.3 3 1025 251.5 3 1026 97 3 1025 115.2 3 1026 10.8 3 1025 4.41 3 1026 12.8 3 1025 5.63 3 1026 12.7 mm 379 mm 15.2 mm Driving distance 575 mm PZ-27 9.53 mm 304.8 mm 9.53 mm Driving distance 457.2 mm PZ-22 9.53 mm 228.6 mm 9.53 mm Driving distance PSA-31 12.7 mm Driving distance PSA-23 558.8 mm 500 mm 9.53 mm Driving distance 406.4 mm 9.2 Construction Methods 9.2 441 Construction Methods Sheet pile walls may be divided into two basic categories: (a) cantilever and (b) anchored. In the construction of sheet pile walls, the sheet pile may be driven into the ground and then the backfill placed on the land side, or the sheet pile may first be driven into the ground and the soil in front of the sheet pile dredged. In either case, the soil used for backfill behind the sheet pile wall is usually granular. The soil below the dredge line may be sandy or clayey. The surface of soil on the water side is referred to as the mud line or dredge line. Thus, construction methods generally can be divided into two categories (Tsinker, 1983): 1. Backfilled structure 2. Dredged structure The sequence of construction for a backfilled structure is as follows (see Figure 9.5): Step 1. Step 2. Step 3. Step 4. Dredge the in situ soil in front and back of the proposed structure. Drive the sheet piles. Backfill up to the level of the anchor, and place the anchor system. Backfill up to the top of the wall. For a cantilever type of wall, only Steps 1, 2, and 4 apply. The sequence of construction for a dredged structure is as follows (see Figure 9.6): Step 1. Step 2. Step 3. Step 4. Drive the sheet piles. Backfill up to the anchor level, and place the anchor system. Backfill up to the top of the wall. Dredge the front side of the wall. With cantilever sheet pile walls, Step 2 is not required. Original ground surface Dredge Step 1 Dredge line Step 2 Anchor rod Backfill Step 3 Backfill Step 4 Figure 9.5 Sequence of construction for a backfilled structure 442 Chapter 9: Sheet Pile Walls Anchor rod Original ground surface Backfill Step 1 Step 2 Backfill Dredge Step 3 9.3 Step 4 Figure 9.6 Sequence of construction for a dredged structure Cantilever Sheet Pile Walls Cantilever sheet pile walls are usually recommended for walls of moderate height— about 6 m or less, measured above the dredge line. In such walls, the sheet piles act as a wide cantilever beam above the dredge line. The basic principles for estimating net lateral pressure distribution on a cantilever sheet-pile wall can be explained with the aid of Figure 9.7. The figure shows the nature of lateral yielding of a cantilever wall penetrating a sand layer below the dredge line. The wall rotates about point O (Figure 9.7a). Because the hydrostatic pressures at any depth from both sides of the wall will cancel each other, we consider only the effective lateral soil pressures. In zone A, the lateral pressure is just the active pressure from the land side. In zone B, because of the nature of yielding of the wall, there will be active pressure from the land side and passive pressure from the water side. The condition is reversed in zone C—that is, below the point of rotation, O. The net actual pressure distribution on the wall is like that shown in Figure 9.7b. However, for design purposes, Figure 9.7c shows a simplified version. Sections 9.4 through 9.7 present the mathematical formulation of the analysis of cantilever sheet pile walls. Note that, in some waterfront structures, the water level may fluctuate as the result of tidal effects. Care should be taken in determining the water level that will affect the net pressure diagram. 9.4 Cantilever Sheet Piling Penetrating Sandy Soils To develop the relationships for the proper depth of embedment of sheet piles driven into a granular soil, examine Figure 9.8a. The soil retained by the sheet piling above the dredge line also is sand. The water table is at a depth L1 below the top of the wall. 9.4 Cantilever Sheet Piling Penetrating Sandy Soils Water table Zone A Active pressure Sand Dredge line Passive pressure Active pressure O Zone B Active pressure Passive pressure Zone C Sand (a) (b) (c) Figure 9.7 Cantilever sheet pile penetrating sand A Water table Sand ␥ c = 0 L1 1 C Sand ␥sat c = 0 L L2 z P 2 Dredge line D Slope: 1 vertical: (Kp – Ka)␥ horizontal L3 E D F L4 F z F L5 H 3 B (a) z 4 Mmax Sand ␥sat G c = 0 (b) Figure 9.8 Cantilever sheet pile penetrating sand: (a) variation of net pressure diagram; (b) variation of moment 443 444 Chapter 9: Sheet Pile Walls Let the effective angle of friction of the sand be fr. The intensity of the active pressure at a depth z 5 L1 is s1r 5 gL1Ka (9.1) where Ka 5 Rankine active pressure coefficient 5 tan2 (45 2 fr>2) g 5 unit weight of soil above the water table Similarly, the active pressure at a depth z 5 L1 1 L2 (i.e., at the level of the dredge line) is s2r 5 (gL1 1 grL2 )Ka (9.2) where gr 5 effective unit weight of soil 5 gsat 2 gw. Note that, at the level of the dredge line, the hydrostatic pressures from both sides of the wall are the same magnitude and cancel each other. To determine the net lateral pressure below the dredge line up to the point of rotation, O, as shown in Figure 9.7a, an engineer has to consider the passive pressure acting from the left side (the water side) toward the right side (the land side) of the wall and also the active pressure acting from the right side toward the left side of the wall. For such cases, ignoring the hydrostatic pressure from both sides of the wall, the active pressure at depth z is sar 5 3gL1 1 grL2 1 gr(z 2 L1 2 L2 )4Ka (9.3) Also, the passive pressure at depth z is spr 5 gr(z 2 L1 2 L2 )Kp (9.4) where Kp 5 Rankine passive pressure coefficient 5 tan2 (45 1 fr>2) . Combining Eqs. (9.3) and (9.4) yields the net lateral pressure, namely, sr 5 sar 2 spr 5 (gL1 1 grL2 )Ka 2 gr (z 2 L1 2 L2 ) (Kp 2 Ka ) 5 sr2 2 gr(z 2 L) (Kp 2 Ka ) (9.5) where L 5 L1 1 L2. The net pressure, sr equals zero at a depth L3 below the dredge line, so s2r 2 gr(z 2 L) (Kp 2 Ka ) 5 0 or (z 2 L) 5 L3 5 s2r gr(Kp 2 Ka ) (9.6) 445 9.4 Cantilever Sheet Piling Penetrating Sandy Soils Equation (9.6) indicates that the slope of the net pressure distribution line DEF is 1 vertical to (Kp 2 Ka )gr horizontal, so, in the pressure diagram, HB 5 s3r 5 L4 (Kp 2 Ka )gr (9.7) At the bottom of the sheet pile, passive pressure, spr , acts from the right toward the left side, and active pressure acts from the left toward the right side of the sheet pile, so, at z 5 L 1 D, spr 5 (gL1 1 grL2 1 grD)Kp (9.8) sar 5 grDKa (9.9) At the same depth, Hence, the net lateral pressure at the bottom of the sheet pile is spr 2 sar 5 s4r 5 (gL1 1 grL2 )Kp 1 grD(Kp 2 Ka ) 5 (gL1 1 grL2 )Kp 1 grL3 (Kp 2 Ka ) 1 grL4 (Kp 2 Ka ) (9.10) 5 s5r 1 grL4 (Kp 2 Ka ) where s5r 5 (gL1 1 grL2 )Kp 1 grL3 (Kp 2 Ka ) D 5 L3 1 L 4 (9.11) (9.12) For the stability of the wall, the principles of statics can now be applied: S horizontal forces per unit length of wall 5 0 and S moment of the forces per unit length of wall about point B 5 0 For the summation of the horizontal forces, we have Area of the pressure diagram ACDE 2 area of EFHB 1 area of FHBG 5 0 or P 2 12s3r L4 1 12L5 (s3r 1 s4r ) 5 0 (9.13) where P 5 area of the pressure diagram ACDE. Summing the moment of all the forces about point B yields L4 L5 1 1 P(L4 1 z) 2 ¢ L4s3r ≤ ¢ ≤ 1 L5 (s3r 1 s4r ) ¢ ≤ 5 0 2 3 2 3 (9.14) From Eq. (9.13), L5 5 s3r L4 2 2P s3r 1 s4r (9.15) 446 Chapter 9: Sheet Pile Walls Combining Eqs. (9.7), (9.10), (9.14), and (9.15) and simplifying them further, we obtain the following fourth-degree equation in terms of L4 : L44 1 A 1L34 2 A 2L24 2 A 3L4 2 A 4 5 0 (9.16) In this equation, A1 5 s5r gr(Kp 2 Ka ) (9.17) A2 5 8P gr(Kp 2 Ka ) (9.18) A3 5 A4 5 6P32zgr(Kp 2 Ka ) 1 s5r 4 gr2 (Kp 2 Ka ) 2 P(6zs5r 1 4P) gr2 (Kp 2 Ka ) 2 (9.19) (9.20) Step-by-Step Procedure for Obtaining the Pressure Diagram Based on the preceding theory, a step-by-step procedure for obtaining the pressure diagram for a cantilever sheet pile wall penetrating a granular soil is as follows: Step 1. Step 2. Step 3. Step 4. Step 5. Step 6. Step 7. Step 8. Step 9. Step 10. Step 11. Step 12. Step 13. Calculate Ka and Kp . Calculate s1r [Eq. (9.1)] and s2r [Eq. (9.2)]. (Note: L1 and L2 will be given.) Calculate L3 [Eq. (9.6)]. Calculate P. Calculate z (i.e., the center of pressure for the area ACDE) by taking the moment about E. Calculate s5r [Eq. (9.11)]. Calculate A 1 , A 2 , A 3 , and A 4 [Eqs. (9.17) through (9.20)]. Solve Eq. (9.16) by trial and error to determine L4 . Calculate s4r [Eq. (9.10)]. Calculate s3r [Eq. (9.7)]. Obtain L5 from Eq. (9.15). Draw a pressure distribution diagram like the one shown in Figure 9.8a. Obtain the theoretical depth [see Eq. (9.12)] of penetration as L3 1 L4 . The actual depth of penetration is increased by about 20 to 30%. Note that some designers prefer to use a factor of safety on the passive earth pressure coefficient at the beginning. In that case, in Step 1, Kp(design) 5 Kp FS where FS 5 factor of safety (usually between 1.5 and 2). 9.4 Cantilever Sheet Piling Penetrating Sandy Soils 447 For this type of analysis, follow Steps 1 through 12 with the value of Ka 5 tan2 (45 2 fr>2) and Kp(design) (instead of Kp). The actual depth of penetration can now be determined by adding L3 , obtained from Step 3, and L4 , obtained from Step 8. Calculation of Maximum Bending Moment The nature of the variation of the moment diagram for a cantilever sheet pile wall is shown in Figure 9.8b. The maximum moment will occur between points E and Fr. Obtaining the maximum moment (Mmax ) per unit length of the wall requires determining the point of zero shear. For a new axis zr (with origin at point E) for zero shear, P 5 12 (zr) 2 (Kp 2 Ka )gr or zr 5 2P Å (Kp 2 Ka )gr (9.21) Once the point of zero shear force is determined (point Fs in Figure 9.8a), the magnitude of the maximum moment can be obtained as Mmax 5 P(z 1 zr) 2 312 grzr2 (Kp 2 Ka )4 ( 13 )zr (9.22) The necessary profile of the sheet piling is then sized according to the allowable flexural stress of the sheet pile material, or S5 Mmax sall (9.23) where S 5 section modulus of the sheet pile required per unit length of the structure sall 5 allowable flexural stress of the sheet pile Example 9.1 Figure 9.9 shows a cantilever sheet pile wall penetrating a granular soil. Here, L1 ⫽ 2 m, L2 ⫽ 3 m, ␥ ⫽ 15.9 kN/m3, ␥sat ⫽ 19.33 kN/m3, and ⬘ ⫽ 32°. a. What is the theoretical depth of embedment, D? b. For a 30% increase in D, what should be the total length of the sheet piles? c. What should be the minimum section modulus of the sheet piles? Use all ⫽ 172 MN/m2. Solution Part a Using Figure 9.8a for the pressure distribution diagram, one can now prepare the following table for a step-by-step calculation. 448 Chapter 9: Sheet Pile Walls L1 Sand ␥ c = 0 L2 Sand ␥sat c = 0 D Sand ␥sat c = 0 Water table Dredge line Figure 9.9 Cantilever sheet-pile wall Quantity required Eq. no. Ka — Kp — 1⬘ 9.1 ␥L1Ka ⫽ (15.9)(2)(0.307) ⫽ 9.763 kN/m2 2⬘ 9.2 L3 9.6 (␥L1 ⫹ ␥⬘L2)Ka ⫽ [(15.9)(2) ⫹ (19.33 ⫺ 9.81)(3)](0.307) ⫽ 18.53 kN/m2 s2r 18.53 5 5 0.66 m gr (Kp 2 Ka ) (19.33 2 9.81) (3.25 2 0.307) P — Equation and calculation fr 32 b 5 tan2 a45 2 b 5 0.307 2 2 r f 32 tan2 a45 1 b 5 tan2 a45 1 b 5 3.25 2 2 tan2 a45 2 1 1 1 2 s1r L1 1 s1r L2 1 2 (s2r 2 s1r )L2 1 2 s2r L3 1 5 ( 2 ) (9.763) (2) 1 (9.763) (3) 1 ( 12 ) (18.53 1 ( 12 ) (18.53) (0.66) 2 9.763) (3) 5 9.763 1 29.289 1 13.151 1 6.115 5 58.32 kN>m z — 5⬘ 9.11 A1 9.17 A2 9.18 SME 1 9.763(0.66 1 3 1 23 ) 1 29.289(0.66 1 32 ) 5 C S 5 2.23 m P 58.32 1 13.151(0.66 1 33 ) 1 6.115(0.66 3 23 ) (␥L1 ⫹ ␥⬘L2)Kp ⫹ ␥⬘L3(Kp ⫺ Ka) ⫽ [(15.9)(2) ⫹ (19.33 ⫺ 9.81)(3)](3.25) ⫹ (19.33 ⫺ 9.81)(0.66)(3.25 ⫺ 0.307) ⫽ 214.66 kN/m2 s5r 214.66 5 5 7.66 gr (Kp 2 Ka ) (19.33 2 9.81) (3.25 2 0.307) (8) (58.32) 8P 5 5 16.65 gr (Kp 2 Ka ) (19.33 2 9.81) (3.25 2 0.307) 9.5 Special Cases for Cantilever Walls Penetrating a Sandy Soil A3 9.19 6P32zgr (Kp 2 Ka ) 1 s5r 4 5 gr2 (Kp 2 Ka ) 2 (6) (58.32) 3 (2) (2.23) (19.33 2 9.81) (3.25 2 0.307) 1 214.664 (19.33 2 9.81) 2 (3.25 2 0.307) 2 5 151.93 A4 9.20 449 P(6zs5r 1 4P) 2 gr (Kp 2 Ka ) 2 5 58.323(6) (2.23) (214.66) 1 (4) (58.32) 4 (19.33 2 9.81) 2 (3.25 2 0.307) 2 5 230.72 L4 9.16 L44 ⫹ A1L43 ⫺ A2L42 ⫺ A3L4 ⫺ A4 ⫽ 0 L44 ⫹ 7.66L43 ⫺ 16.65L42 ⫺ 151.93L4 ⫺ 230.72 ⫽ 0; L4 ⬇ 4.8 m Thus, Dtheory ⫽ L3 ⫹ L4 ⫽ 0.66 ⫹ 4.8 ⫽ 5.46 m Part b The total length of the sheet piles is L1 ⫹ L2 ⫹ 1.3(L3 ⫹ L4) ⫽ 2 ⫹ 3 ⫹ 1.3(5.46) ⫽ 12.1 m Part c Finally, we have the following table. Quantity required Eq. no. z⬘ 9.21 Mmax 9.22 Equation and calculation (2) (58.32) 2P 5 5 2.04 m Ä (Kp 2 Ka )gr Å (3.25 2 0.307) (19.33 2 9.81) 1 zr P(z 1 z r ) 2 c grz r2 (Kp 2 Ka ) d 5 (58.32) (2.23 1 2.04) 2 3 1 2.04 2 c a b (19.33 2 9.81) (2.04) 2 (3.25 2 0.307) d 2 3 5 209.39 kN # m>m S 9.5 9.29 Mmax 209.39 kN # m 5 5 1.217 3 1023 m3>m of wall s all 172 3 103 kN>m2 ■ Special Cases for Cantilever Walls Penetrating a Sandy Soil Sheet Pile Wall with the Absence of Water Table In the absence of the water table, the net pressure diagram on the cantilever sheet-pile wall will be as shown in Figure 9.10, which is a modified version of Figure 9.8. In this case, 450 Chapter 9: Sheet Pile Walls Sand ␥ L P z 2 Sand ␥ L3 D L4 Figure 9.10 Sheet piling penetrating a sandy soil in the absence of the water table L5 3 4 s2r 5 gLKa s3r 5 L4 (Kp 2 Ka )g (9.24) (9.25) s4r 5 s5r 1 gL4 (Kp 2 Ka ) (9.26) s5r 5 gLKp 1 gL3 (Kp 2 Ka ) (9.27) L3 5 s2r LKa 5 g(Kp 2 Ka ) (Kp 2 Ka ) P 5 12s2r L 1 12s2r L3 z 5 L3 1 (9.28) (9.29) L(2Ka 1 Kp ) LKa L L 5 1 5 3 Kp 2 Ka 3 3(Kp 2 Ka ) (9.30) and Eq. (9.16) transforms to L44 1 A 1r L34 2 A 2r L24 2 A 3r L4 2 A 4r 5 0 (9.31) where A 1r 5 s5r g(Kp 2 Ka ) (9.32) A 2r 5 8P g(Kp 2 Ka ) (9.33) A 3r 5 A 4r 5 6P32zg(Kp 2 Ka ) 1 s5r 4 g2 (Kp 2 Ka ) 2 P(6zs5r 1 4P) g2 (Kp 2 Ka ) 2 (9.34) (9.35) 451 9.5 Special Cases for Cantilever Walls Penetrating a Sandy Soil P L Sand ␥ c = 0 D L5 3 = ␥D (Kp – Ka) 4 = ␥D (Kp – Ka) Figure 9.11 Free cantilever sheet piling penetrating a layer of sand Free Cantilever Sheet Piling Figure 9.11 shows a free cantilever sheet-pile wall penetrating a sandy soil and subjected to a line load of P per unit length of the wall. For this case, D4 2 c 2 8P 12PL 2P d D2 2 c dD 2 c d 50 g(Kp 2 Ka ) g(Kp 2 Ka ) g(Kp 2 Ka ) L5 5 g(Kp 2 Ka )D2 2 2P 2D(Kp 2 Ka )g Mmax 5 P(L 1 z r ) 2 gz93 (Kp 2 Ka ) 6 (9.36) (9.37) (9.38) and zr 5 2P Å gr (Kp 2 Ka ) (9.39) Example 9.2 Redo parts a and b of Example 9.1, assuming the absence of the water table. Use ␥ ⫽ 15.9 kN/m3 and ⬘ ⫽ 32°. Note: L ⫽ 5 m. 452 Chapter 9: Sheet Pile Walls Solution Part a Quantity required Eq. no. Equation and calculation Ka — tan2 a45 2 fr 32 b 5 tan2 a45 2 b 5 0.307 2 2 Kp — tan2 a45 1 fr 32 b 5 tan2 a45 1 b 5 3.25 2 2 2⬘ 9.24 ␥LKa ⫽ (15.9)(5)(0.307) ⫽ 24.41 kN/m2 L3 9.28 (5) (0.307) LKa 5 5 0.521 m Kp 2 Ka 3.25 2 0.307 5⬘ 9.27 ␥LKp ⫹ ␥L3(Kp ⫺ Ka) ⫽ (15.9)(5)(3.25) ⫹ (15.9)(0.521)(3.25 ⫺ 0.307) ⫽ 282.76 kN/m2 P 9.29 1 2 z 9.30 A1⬘ 9.32 A2⬘ 9.33 A3⬘ 9.34 s2r L 1 12 s2r L3 5 12 s2r (L 1 L3 ) 5 ( 12 ) (24.41) (5 1 0.521) 5 67.38 kN>m L(2Ka 2 Kp ) 9.35 L4 9.31 53 (2) (0.307) 1 3.254 5 2.188 m 3(Kp 2 Ka ) 3(3.25 2 0.307) s5r 282.76 5 5 6.04 g(Kp 2 Ka ) (15.9) (3.25 2 0.307) (8) (67.38) 8P 5 5 11.52 g(Kp 2 Ka ) (15.9) (3.25 2 0.307) 6P32zg(Kp 2 Ka ) 1 sr54 g2 (Kp 2 Ka ) 2 5 A4⬘ 5 (6) (67.38) 3(2) (2.188) (15.9) (3.25 2 0.307) 1 282.764 (15.9) 2 (3.25 2 0.307) 2 P(6zs5r 1 4P) 2 g (Kp 2 Ka ) 2 5 5 90.01 (67.38) 3(6) (2.188) (282.76) 1 (4) (67.38)4 (15.9) 2 (3.25 2 0.307) 2 5 122.52 L44 ⫹ A1⬘L43 ⫺ A2⬘L42 ⫺ A3⬘L4 ⫺ A4⬘ ⫽ 0 L44 ⫹ 6.04L43 ⫺ 11.52L24 ⫺ 90.01L4 ⫺ 122.52 ⫽ 0; L4 ⬇ 4.1 m Dtheory ⫽ L3 ⫹ L4 ⫽ 0.521 ⫹ 4.1 ⬇ 4.7 m Part b Total length, L ⫹ 1.3(Dtheory) ⫽ 5 ⫹ 1.3(4.7) ⫽ 11.11 m 9.6 ■ Cantilever Sheet Piling Penetrating Clay At times, cantilever sheet piles must be driven into a clay layer possessing an undrained cohesion c(f 5 0). The net pressure diagram will be somewhat different from that shown in Figure 9.8a. Figure 9.12 shows a cantilever sheet-pile wall driven into clay with a backfill of granular soil above the level of the dredge line. The water table is at 9.6 Cantilever Sheet Piling Penetrating Clay 453 A Sand ␥ c=0 L1 Water table 1 C Sand ␥sat c=0 z L2 P1 z1 Dredge line F 2 E D 6 L3 Clay ␥sat =0 c G D z L4 I B 7 H Figure 9.12 Cantilever sheet pile penetrating clay a depth L1 below the top of the wall. As before, Eqs. (9.1) and (9.2) give the intensity of the net pressures s1r and s2r , and the diagram for pressure distribution above the level of the dredge line can be drawn. The diagram for net pressure distribution below the dredge line can now be determined as follows. At any depth greater than L1 1 L2 , for f 5 0, the Rankine active earth-pressure coefficient Ka 5 1. Similarly, for f 5 0, the Rankine passive earth-pressure coefficient Kp 5 1. Thus, above the point of rotation (point O in Figure 9.7a), the active pressure, from right to left is sa 5 3gL1 1 grL2 1 gsat (z 2 L1 2 L2 )4 2 2c (9.40) Similarly, the passive pressure from left to right may be expressed as sp 5 gsat (z 2 L1 2 L2 ) 1 2c (9.41) Thus, the net pressure is s6 5 sp 2 sa 5 3gsat (z 2 L1 2 L2 ) 1 2c4 2 3gL1 1 grL2 1 gsat (z 2 L1 2 L2 ) 4 1 2c 5 4c 2 (gL1 1 grL2 ) (9.42) At the bottom of the sheet pile, the passive pressure from right to left is sp 5 (gL1 1 grL2 1 gsatD) 1 2c (9.43) 454 Chapter 9: Sheet Pile Walls Similarly, the active pressure from left to right is sa 5 gsatD 2 2c (9.44) s7 5 sp 2 sa 5 4c 1 (gL1 1 grL2 ) (9.45) Hence, the net pressure is For equilibrium analysis, SFH 5 0; that is, the area of the pressure diagram ACDE minus the area of EFIB plus the area of GIH 5 0, or P1 2 34c 2 (gL1 1 grL2 ) 4D 1 12L4 34c 2 (gL1 1 grL2 ) 1 4c 1 (gL1 1 grL2 )4 5 0 where P1 5 area of the pressure diagram ACDE. Simplifying the preceding equation produces L4 5 D34c 2 (gL1 1 grL2 ) 4 2 P1 4c (9.46) Now, taking the moment about point B (SMB 5 0) yields P1 (D 1 z1 ) 2 34c 2 (gL1 1 grL2 )4 L4 D2 1 1 L4 (8c) ¢ ≤ 5 0 2 2 3 (9.47) where z1 5 distance of the center of pressure of the pressure diagram ACDE, measured from the level of the dredge line. Combining Eqs. (9.46) and (9.47) yields D2 34c 2 (gL1 1 grL2 ) 4 2 2DP1 2 P1 (P1 1 12cz1 ) 50 (gL1 1 grL2 ) 1 2c (9.48) Equation (9.48) may be solved to obtain D, the theoretical depth of penetration of the clay layer by the sheet pile. Step-by-Step Procedure for Obtaining the Pressure Diagram Step 1. Calculate Ka 5 tan2 (45 2 fr>2) for the granular soil (backfill). Step 2. Obtain s1r and s2r . [See Eqs. (9.1) and (9.2).] Step 3. Calculate P1 and z1 . Step 4. Use Eq. (9.48) to obtain the theoretical value of D. Step 5. Using Eq. (9.46), calculate L4 . Step 6. Calculate s6 and s7 . [See Eqs. (9.42) and (9.45).] Step 7. Draw the pressure distribution diagram as shown in Figure 9.12. Step 8. The actual depth of penetration is Dactual 5 1.4 to 1.6(Dtheoretical ) 9.6 Cantilever Sheet Piling Penetrating Clay 455 Maximum Bending Moment According to Figure 9.12, the maximum moment (zero shear) will be between L1 1 L2 , z , L1 1 L2 1 L3 . Using a new coordinate system zr (with zr 5 0 at the dredge line) for zero shear gives P1 2 s6zr 5 0 or zr 5 P1 s6 (9.49) The magnitude of the maximum moment may now be obtained: Mmax 5 P1 (zr 1 z1 ) 2 s6zr2 2 (9.50) Knowing the maximum bending moment, we determine the section modulus of the sheet pile section from Eq. (9.23). Example 9.3 In Figure 9.13, for the sheet pile wall, determine a. The theoretical and actual depth of penetration. Use Dactual 5 1.5Dtheory . b. The minimum size of sheet pile section necessary. Use sall 5 172.5 MN>m2. A L1 = 2 m Sand ␥ = 15.9 kN/m3 c = 0 = 32° L2 = 3 m Sand ␥sat = 19.33 kN/m3 c = 0 = 32° D Clay c = 47 kN/m2 =0 Water table E B Figure 9.13 Cantilever sheet pile penetrating into saturated clay 456 Chapter 9: Sheet Pile Walls Solution We will follow the step-by-step procedure given in Section 9.6: Step 1. Ka 5 tan2 ¢ 45 2 fr 32 ≤ 5 tan2 ¢45 2 ≤ 5 0.307 2 2 Step 2. s1r 5 gL1Ka 5 (15.9) (2) (0.307) 5 9.763 kN>m2 s2r 5 (gL1 1 grL2 )Ka 5 3(15.9) (2) 1 (19.33 2 9.81)340.307 5 18.53 kN>m2 Step 3. From the net pressure distribution diagram given in Figure 9.12, we have 1 1 P1 5 s1r L1 1 s1r L2 1 (s2r 2 s1r )L2 2 2 5 9.763 1 29.289 1 13.151 5 52.2 kN>m and 1 2 3 3 B9.763¢3 1 ≤ 1 29.289¢ ≤ 1 13.151¢ ≤ R 52.2 3 2 3 5 1.78 m z1 5 Step 4. From Eq. (9.48), D2 34c 2 (gL1 1 grL2 )4 2 2DP1 2 P1 (P1 1 12cz1 ) 50 (gL1 1 grL2 ) 1 2c Substituting proper values yields D2 5 (4) (47) 2 3(2) (15.9) 1 (19.33 2 9.81)346 2 2D(52.2) 52.2352.2 1 (12) (47) (1.78)4 2 50 3(15.9) (2) 1 (19.33 2 9.81)34 1 (2) (47) or 127.64D2 2 104.4D 2 357.15 5 0 Step 5. Solving the preceding equation, we obtain D 5 2.13 m. From Eq. (9.46), L4 5 D34c 2 (gL1 1 grL2 ) 4 2 P1 4c and 4c 2 (gL1 1 grL2 ) 5 (4) (47) 2 3(15.9) (2) 1 (19.33 2 9.81)34 5 127.64 kN>m2 457 9.7 Special Cases for Cantilever Walls Penetrating Clay So, L4 5 2.13(127.64) 2 52.2 5 1.17 m (4) (47) Step 6. s6 5 4c 2 (gL1 1 grL2 ) 5 127.64 kN>m2 s7 5 4c 1 (gL1 1 grL2 ) 5 248.36 kN>m2 Step 7. Step 8. The net pressure distribution diagram can now be drawn, as shown in Figure 9.12. Dactual < 1.5Dtheoretical 5 1.5(2.13) < 3.2 m Maximum-Moment Calculation From Eq. (9.49), zr 5 P1 52.2 5 < 0.41 m s6 127.64 Again, from Eq. (9.50), Mmax 5 P1 (zr 1 z1 ) 2 s6zr2 2 So 127.64(0.41) 2 2 5 114.32 2 10.73 5 103.59 kN-m>m Mmax 5 52.2(0.41 1 1.78) 2 The minimum required section modulus (assuming that sall 5 172.5 MN>m2) is S5 9.7 103.59 kN-m>m 5 0.6 3 1023 m3 , m of the wall 172.5 3 103 kN>m2 ■ Special Cases for Cantilever Walls Penetrating Clay Sheet Pile Wall in the Absence of Water Table As in Section 9.5, relationships for special cases for cantilever walls penetrating clay may also be derived. Referring to Figure 9.14, we can write s2r 5 gLKa (9.51) s6 5 4c 2 gL (9.52) s7 5 4c 1 gL (9.53) P1 5 1 2 Ls2r 1 2 2 gL Ka (9.54) L4 5 D(4c 2 gL) 2 12gL2Ka 4c (9.55) 5 and 458 Chapter 9: Sheet Pile Walls Sand ␥ L P1 z1 2 6 Clay L3 ␥sat = 0 c D L4 7 Figure 9.14 Sheet-pile wall penetrating clay The theoretical depth of penetration, D, can be calculated [in a manner similar to the calculation of Eq. (9.48)] as D2 (4c 2 gL) 2 2DP1 2 where z1 5 P1 (P1 1 12cz1 ) 50 gL 1 2c L . 3 (9.56) (9.57) The magnitude of the maximum moment in the wall is Mmax 5 P1 (zr 1 z1 ) 2 where zr 5 s6zr2 2 1 2 P1 2 gL Ka . 5 s6 4c 2 gL (9.58) (9.59) Free Cantilever Sheet-Pile Wall Penetrating Clay Figure 9.15 shows a free cantilever sheet-pile wall penetrating a clay layer. The wall is being subjected to a line load of P per unit length. For this case, 6 ⫽ 7 ⫽ 4c (9.60) The depth of penetration, D, may be obtained from the relation 4D2c 2 2PD 2 P(P 1 12cL) 50 2c (9.61) 9.7 Special Cases for Cantilever Walls Penetrating Clay 459 P L 6 L3 Clay D L4 ␥sat = 0 c 7 Figure 9.15 Free cantilever sheet piling penetrating clay Also, note that, for a construction of the pressure diagram, 4cD 2 P 4c The maximum moment in the wall is L4 5 Mmax 5 P(L 1 z r ) 2 where zr 5 (9.62) 4cz r2 2 P 4c (9.63) (9.64) Example 9.4 Refer to the free cantilever sheet-pile wall shown in Figure 9.15, for which P ⫽ 32 kN/m, L ⫽ 3.5 m, and c ⫽ 12 kN/m2. Calculate the theoretical depth of penetration. Solution From Eq. (9.61), P(P 1 12cL) 50 2c 32332 1 (12) (12) (3.5)4 (4) (D2 ) (12) 2 (2) (32) (D) 2 50 (2) (12) 48D2 ⫺ 64D ⫺ 714.7 ⫽ 0 4D2c 2 2PD 2 Hence D ⬇ 4.6 m. ■ 460 Chapter 9: Sheet Pile Walls 9.8 Anchored Sheet-Pile Walls When the height of the backfill material behind a cantilever sheet-pile wall exceeds about 6 m, tying the wall near the top to anchor plates, anchor walls, or anchor piles becomes more economical. This type of construction is referred to as anchored sheetpile wall or an anchored bulkhead. Anchors minimize the depth of penetration required by the sheet piles and also reduce the cross-sectional area and weight of the sheet piles needed for construction. However, the tie rods and anchors must be carefully designed. The two basic methods of designing anchored sheet-pile walls are (a) the free earth support method and (b) the fixed earth support method. Figure 9.16 shows the assumed nature of deflection of the sheet piles for the two methods. Anchor tie rod Water table Moment Mmax Dredge line D Sheet pile simply supported (a) Anchor tie rod Moment Water table Mmax Deflection Dredge line Point of inflection D Sheet pile fixed at lower end (b) Figure 9.16 Nature of variation of deflection and moment for anchored sheet piles: (a) free earth support method; (b) fixed earth support method 9.9 Free Earth Support Method for Penetration of Sandy Soil 461 The free earth support method involves a minimum penetration depth. Below the dredge line, no pivot point exists for the static system. The nature of the variation of the bending moment with depth for both methods is also shown in Figure 9.16. Note that Dfree earth , Dfixed earth 9.9 Free Earth Support Method for Penetration of Sandy Soil Figure 9.17 shows an anchor sheet-pile wall with a granular soil backfill; the wall has been driven into a granular soil. The tie rod connecting the sheet pile and the anchor is located at a depth l1 below the top of the sheet-pile wall. The diagram of the net pressure distribution above the dredge line is similar to that shown in Figure 9.8. At depth z 5 L1 , s1r 5 gL1Ka , and at z 5 L1 1 L2 , s2r 5 (gL1 1 grL2 )Ka . Below the dredge line, the net pressure will be zero at z 5 L1 1 L2 1 L3 . The relation for L3 is given by Eq. (9.6), or L3 5 s2r gr(Kp 2 Ka ) A Anchor tie rod L1 O Water table 1 Water table C l1 F Sand ␥, l2 z L2 P z 2 Dredge line L3 D 1 E D ␥(Kp – Ka) L4 F 8 Sand ␥sat, B Figure 9.17 Anchored sheet-pile wall penetrating sand Sand ␥sat, 462 Chapter 9: Sheet Pile Walls At z 5 L1 1 L2 1 L3 1 L4 , the net pressure is given by s8r 5 gr(Kp 2 Ka )L4 (9.65) Note that the slope of the line DEF is 1 vertical to gr(Kp 2 Ka ) horizontal. For equilibrium of the sheet pile, S horizontal forces 5 0, and S moment about Or 5 0. (Note: Point Or is located at the level of the tie rod.) Summing the forces in the horizontal direction (per unit length of the wall) gives Area of the pressure diagram ACDE 2 area of EBF 2 F 5 0 where F 5 tension in the tie rod> unit length of the wall, or P 2 12 s8r L4 2 F 5 0 or F 5 P 2 12 3 gr(Kp 2 Ka )4L24 (9.66) where P 5 area of the pressure diagram ACDE. Now, taking the moment about point Or gives 2P3(L1 1 L2 1 L3 ) 2 (z 1 l1 )4 1 12 3gr(Kp 2 Ka )4L24 (l2 1 L2 1 L3 1 23L4 ) 5 0 or L34 1 1.5L24 (l2 1 L2 1 L3 ) 2 3P3(L1 1 L2 1 L3 ) 2 (z 1 l1 )4 gr (Kp 2 Ka ) 50 (9.67) Equation (9.67) may be solved by trial and error to determine the theoretical depth, L4 : Dtheoretical 5 L3 1 L4 The theoretical depth is increased by about 30 to 40% for actual construction, or Dactual 5 1.3 to 1.4Dtheoretical (9.68) The step-by-step procedure in Section 9.4 indicated that a factor of safety can be applied to Kp at the beginning [i.e., Kp(design) 5 Kp>FS]. If that is done, there is no need to increase the theoretical depth by 30 to 40%. This approach is often more conservative. 9.9 Free Earth Support Method for Penetration of Sandy Soil 463 The maximum theoretical moment to which the sheet pile will be subjected occurs at a depth between z 5 L1 and z 5 L1 1 L2 . The depth z for zero shear and hence maximum moment may be evaluated from 1 2 s1r L1 2 F 1 s1r (z 2 L1 ) 1 12Kagr(z 2 L1 ) 2 5 0 (9.69) Once the value of z is determined, the magnitude of the maximum moment is easily obtained. Example 9.5 Let L1 ⫽ 3.05 m, L2 ⫽ 6.1 m, l1 ⫽ 1.53 m, l2 ⫽ 1.52 m, c⬘ ⫽ 0, ⬘ ⫽ 30°, ␥ ⫽ 16 kN/m3, ␥sat ⫽ 19.5 kN/m3, and E ⫽ 207 ⫻ 103 MN/m2 in Figure 9.17. a. Determine the theoretical and actual depths of penetration. (Note: Dactual ⫽ 1.3Dtheory.) b. Find the anchor force per unit length of the wall. c. Determine the maximum moment, Mmax. Solution Part a We use the following table. Quantity required Eq. no. Ka — tan2 a45 2 fr 30 1 b 5 tan2 a45 2 b 5 2 2 3 KP — tan2 a45 1 fr 30 b 5 tan2 a45 1 b 5 3 2 2 Kp ⫺ Ka ␥⬘ — — 3 ⫺ 0.333 ⫽ 2.667 ␥sat ⫺ ␥w ⫽ 19.5 ⫺ 9.81 ⫽ 9.69 kN/m3 1⬘ 9.1 gL1Ka 5 (16) (3.05) ( 13 ) 5 16.27 kN>m2 2⬘ 9.2 L3 9.6 s2r 35.97 5 5 1.39 m gr (Kp 2 Ka ) (9.69) (2.667) P — 1 2 s1r L1 Equation and calculation (gL1 1 grL2 )Ka 5 3 (16) (3.05) 1 (9.69) (6.1)4 13 5 35.97 kN>m2 1 s2r L2 1 12 (s2r 2 s1r )L2 1 12s2r L3 5 ( 12 ) (16.27) (3.05) 1 (16.27) (6.1) 1 ( 12 ) (35.97 2 16.27) (6.1) 1 ( 12 ) (35.97) (1.39) 5 24.81 1 99.25 1 60.01 1 25.0 5 209.07 kN>m z — 3.05 6.1 b 1 (99.25) a1.39 1 b 3 2 1 ¥ 6.1 2 3 1.39 209.07 1 (60.01) a1.39 1 b 1 (25.0) a b 3 3 5 4.21 m SME 5 ≥ P (24.81) a1.39 1 6.1 1 464 Chapter 9: Sheet Pile Walls L4 L34 1 1.5L24 (l2 1 L2 1 L3 ) 2 9.67 3P3 (L1 1 L2 1 L3 ) 2 (z 1 l1 )4 gr(Kp 2 Ka ) 50 L43 ⫹ 1.5L42(1.52 ⫹ 6.1 ⫹ 1.39) 2 (3) (209.07) 3 (3.05 1 6.1 1 1.39) 2 (4.21 1 1.53)4 (9.69) (2.667) 50 L4 5 2.7 m Dtheory — L3 ⫹ L4 ⫽ 1.39 ⫹ 2.7 ⫽ 4.09 ⬇ 4.1 m Dactual — 1.3Dtheory ⫽ (1.3)(4.1) ⫽ 5.33 m Part b The anchor force per unit length of the wall is F 5 P 2 12gr(Kp 2 Ka )L24 5 209.07 2 A 12 B (9.69) (2.667) (2.7) 2 5 114.87 kN>m < 115 kN>m Part c From Eq. (9.69), for zero shear, 1 2 s1r L1 2 F 1 s1r (z 2 L1 ) 1 12 Kagr (z 2 L1 ) 2 5 0 Let z ⫺ L1 ⫽ x, so that 1 2 s1r l1 2 F 1 s1r x 1 12 Kagrx2 5 0 or A 12 B (16.27) (3.05) 2 115 1 (16.27) (x) 1 A 12 BA 13 B (9.69)x2 5 0 giving x2 ⫹ 10.07x ⫺ 55.84 ⫽ 0 Now, x ⫽ 4 m and z ⫽ x ⫹ L1 ⫽ 4 ⫹ 3.05 ⫽ 7.05 m. Taking the moment about the point of zero shear, we obtain 1 1 3.05 x2 x b 1 F(x 1 1.52) 2 s1r 2 Kagrx2 a b Mmax 5 2 s1r L1 ax 1 2 3 2 2 3 or 1 3.05 42 Mmax 5 2 a b (16.27) (3.05) a4 1 b 1 (115) (4 1 1.52) 2 (16.27) a b 2 3 2 1 1 4 2 a b a b (9.69) (4) 2 a b 5 344.9 kN ? m>m 2 3 3 ■ 9.10 Design Charts for Free Earth Support Method (Penetration into Sandy Soil) 9.10 465 Design Charts for Free Earth Support Method (Penetration into Sandy Soil) Using the free earth support method, Hagerty and Nofal (1992) provided simplified design charts for quick estimation of the depth of penetration, D, anchor force, F, and maximum moment, Mmax, for anchored sheet-pile walls penetrating into sandy soil, as shown in Figure 9.17. They made the following assumptions for their analysis. a. The soil friction angle, fr, above and below the dredge line is the same. b. The angle of friction between the sheet-pile wall and the soil is fr>2. c. The passive earth pressure below the dredge line has a logarithmic spiral failure surface. d. For active earth-pressure calculation, Coulomb’s theory is valid. The magnitudes of D, F, and Mmax may be calculated from the following relationships: D 5 (GD) (CDL1 ) L1 1 L2 (9.70) F 5 (GF) (CFL1 ) ga (L1 1 L2 ) 2 (9.71) Mmax ga (L1 1 L2 ) 3 5 (GM) (CML1 ) (9.72) where ga 5 average unit weight of soil 5 gL21 1 (gsat 2 gw )L22 1 2gL1L2 (L1 1 L2 ) 2 GD 5 generalized nondimensional embedment 5 D L1 1 L2 (for L1 5 0 and L2 5 L1 1 L2 ) GF 5 generalized nondimensional anchor force 5 F ga (L1 1 L2 ) 2 (for L1 5 0 and L2 5 L1 1 L2 ) (9.73) 466 Chapter 9: Sheet Pile Walls GM 5 generalized nondimensional moment 5 Mmax ga (L1 1 L2 ) 3 (for L1 5 0 and L2 5 L1 1 L2 ) CDL1, CFL1, CML1 5 correction factors for L1 2 0 The variations of GD, GF, GM, CDL1, CFL1, and CML1 are shown in Figures 9.18, 9.19, 9.20, 9.21, 9.22, and 9.23, respectively. 0.5 0.4 GD 24 26° 0.3 28° 30° 32° 0.2 34° 36° 38° 0.1 0.0 0.1 0.2 0.3 l1/(L1 L2) 0.4 0.5 Figure 9.18 Variation of GD with l1> (L1 1 L2 ) and fr (Hagerty, D. J., and Nofal, M. M. (1992). “Design Aids: Anchored Bulkheads in Sand,” Canadian Geotechnical Journal, Vol. 29, No. 5, pp. 789–795. © 2008 NRC Canada or its licensors. Reproduced with permission.) 0.16 0.14 24 0.12 26° GF 28° 0.10 30° 32° 34° 36° 38° 0.08 0.06 0.04 0.0 0.1 0.2 0.3 l1/(L1 L2) 0.4 0.5 Figure 9.19 Variation of GF with l1> (L1 1 L2 ) and fr (After Hagerty and Nofal, 1992) (Hagerty, D. J., and Nofal, M. M. (1992). "Design Aids: Anchored Bulkheads in Sand," Canadian Geotechnical Journal, Vol. 29, No. 5, pp. 789-795. © 2008 NRC Canada or its licensors. Reproduced with permission. 0.05 0.04 GM 0.03 24 26° 28° 30° 0.02 0.01 32° 34° 36° 38° 0.00 0.0 0.1 0.2 0.3 l1/(L1 L2) 0.4 0.5 Figure 9.20 Variation of GM with l1> (L1 1 L2 ) and fr (Hagerty, D. J., and Nofal, M. M. (1992). “Design Aids: Anchored Bulkheads in Sand,” Canadian Geotechnical Journal, Vol. 29, No. 5, pp. 789–795. © 2008 NRC Canada or its licensors. Reproduced with permission.) 1.18 1.16 L1 0.4 L1 L2 1.14 CDL1 1.12 0.3 1.10 1.08 0.2 1.06 1.04 0.1 0.0 0.1 0.2 0.3 l1/(L1 L2) 0.4 0.5 Figure 9.21 Variation of CDL1 with L1> (L1 1 L2 ) and l1> (L1 1 L2 ) (Hagerty, D. J., and Nofal, M. M. (1992). “Design Aids: Anchored Bulkheads in Sand,” Canadian Geotechnical Journal, Vol. 29, No. 5, pp. 789–795. © 2008 NRC Canada or its licensors. Reproduced with permission.) 1.08 1.07 L1 0.4 L1 L2 0.3 CFL1 1.06 0.2 1.05 1.04 0.1 1.03 0.0 0.1 0.2 0.3 l1/(L1 L2) 0.4 0.5 Figure 9.22 Variation of CFL1 with L1> (L1 1 L2 ) and l1> (L1 1 L2 ) (Hagerty, D. J., and Nofal, M. M. (1992). “Design Aids: Anchored Bulkheads in Sand,” Canadian Geotechnical Journal, Vol. 29, No. 5, pp. 789–795. © 2008 NRC Canada or its licensors. Reproduced with permission.) 467 468 Chapter 9: Sheet Pile Walls 1.06 1.04 CML1 1.02 1.00 L1 0.4 L1 L2 0.1 0.98 0.3 0.1 0.2 0.96 0.94 0.0 0.1 0.2 0.3 l1/(L1 L2) 0.4 0.5 Figure 9.23 Variation of CML1 with L1> (L1 1 L2 ) and l1> (L1 1 L2 ) (Hagerty, D. J., and Nofal, M. M. (1992). “Design Aids: Anchored Bulkheads in Sand,” Canadian Geotechnical Journal, Vol. 29, No. 5, pp. 789–795. © 2008 NRC Canada or its licensors. Reproduced with permission.) Example 9.6 Refer to Figure 9.17. Given: L1 ⫽ 2 m, L2 ⫽ 3 m, l1 ⫽ l2 ⫽ 1 m, c ⫽ 0, ⬘ ⫽ 32° ␥ ⫽ 15.9 kN/m3, and ␥sat ⫽ 19.33 kN/m3. Determine: a. Theoretical and actual depth of penetration. Note: Dactual ⫽ 1.4Dtheory. b. Anchor force per unit length of wall. c. Maximum moment, Mmax. Use the charts presented in Section 9.10. Solution Part a From Eq. (9.70), D 5 (GD) (CDL1 ) L1 1 L2 l1 1 5 5 0.2 L1 1 L2 213 From Figure 9.18 for l1/(L1 ⫹ L2) ⫽ 0.2 and ⬘ ⫽ 32°, GD ⫽ 0.22. From Figure 9.21, for L1 2 5 5 0.4 L1 1 L2 213 and l1 5 0.2 L1 1 L2 9.11 Moment Reduction for Anchored Sheet-Pile Walls 469 CDL1 ⬇ 1.172. So Dtheory ⫽ (L1 ⫹ L2)(GD)(CDL1) ⫽ (5)(0.22)(1.172) ⬇ 1.3 Dactual ⬇ (1.4)(1.3) ⫽ 1.82 ⬇ 2 m Part b From Figure 9.19 for l1/(L1 ⫹ L2) ⫽ 0.2 and ⬘ ⫽ 32°, GF ⬇ 0.074. Also, from Figure 9.22, for L1 2 5 5 0.4, L1 1 L2 213 l1 5 0.2, L1 1 L2 and fr 5 32° CFL1 ⫽ 1.073. From Eq. (9.73), ga 5 5 gL21 1 g9L22 1 2gL1L2 (L1 1 L2 ) 2 (15.9) (2) 2 1 (19.33 2 9.81) (3) 2 1 (2) (15.9) (2) (3) (2 1 3) 2 5 13.6 kN>m3 Using Eq. (9.71) yields F ⫽ ␥a(L1 ⫹ L2)2(GF)(CFL1) ⫽ (13.6)(5)2(0.074)(1.073) ⬇ 27 kN/m Part c From Figure 9.20, for l1/(L1 ⫹ L2) ⫽ 0.2 and ⫽ 32°, GM ⫽ 0.021. Also, from Figure 9.23, for L1 2 5 5 0.4, L1 1 L2 213 l1 5 0.2, L1 1 L2 and fr 5 32° CML1 ⫽ 1.036. Hence from Eq. (9.72), Mmax ⫽ ␥a(L1 ⫹ L2)3(GM)(CML1) ⫽ (13.6)(5)3(0.021)(1.036) ⫽ 36.99 kN ⭈ m/m 9.11 ■ Moment Reduction for Anchored Sheet-Pile Walls Sheet piles are flexible, and hence sheet-pile walls yield (i.e., become displaced laterally), which redistributes the lateral earth pressure. This change tends to reduce the maximum bending moment, Mmax , as calculated by the procedure outlined in 470 Chapter 9: Sheet Pile Walls 1.0 Loose sand ␣H 0.8 H L1 L2 Dactual Safe section Md Mmax 0.6 Dense sand and gravel 0.4 Unsafe section 0.2 Flexible piles Stiff piles 0 4.0 3.5 3.0 Log ρ 2.5 2.0 Figure 9.24 Plot of log r against Md>Mmax for sheet-pile walls penetrating sand (From Rowe, P. W. (1952). “Anchored Sheet Pile Walls,” Proceedings, Institute of Civil Engineers, Vol. 1, Part 1, pp. 27–70. ) Section 9.9. For that reason, Rowe (1952, 1957) suggested a procedure for reducing the maximum design moment on the sheet pile walls obtained from the free earth support method. This section discusses the procedure of moment reduction for sheet piles penetrating into sand. In Figure 9.24, which is valid for the case of a sheet pile penetrating sand, the following notation is used: 1. Hr 5 total height of pile driven (i.e., L1 1 L2 1 Dactual) 4 2. Hr Relative flexibility of pile 5 r 5 10.91 3 10 ¢ ≤ EI 27 (9.74) where Hr is in meters E 5 modulus of elasticity of the pile material (MN> m2 ) I 5 moment of inertia of the pile section per meter of the wall (m4>m of wall) 3. Md 5 design moment 4. Mmax 5 maximum theoretical moment 9.11 Moment Reduction for Anchored Sheet-Pile Walls 471 The procedure for the use of the moment reduction diagram (see Figure 9.24) is as follows: Step 1. Choose a sheet pile section (e.g., from among those given in Table 9.1). Step 2. Find the modulus S of the selected section (Step 1) per unit length of the wall. Step 3. Determine the moment of inertia of the section (Step 1) per unit length of the wall. Step 4. Obtain Hr and calculate r [see Eq. (9.74)]. Step 5. Find log r. Step 6. Find the moment capacity of the pile section chosen in Step 1 as Md 5 sallS. Step 7. Determine Md>Mmax . Note that Mmax is the maximum theoretical moment determined before. Step 8. Plot log r (Step 5) and Md>Mmax in Figure 9.24. Step 9. Repeat Steps 1 through 8 for several sections. The points that fall above the curve (in loose sand or dense sand, as the case may be) are safe sections. The points that fall below the curve are unsafe sections. The cheapest section may now be chosen from those points which fall above the proper curve. Note that the section chosen will have an Md , Mmax . Example 9.7 Refer to Example 9.5. Use Rowe’s moment reduction diagram (Figure 9.24) to find an appropriate sheet pile section. For the sheet pile, use E 5 207 3 103 MN>m2 and sall 5 172,500 kN>m2. Solution H r 5 L1 1 L2 1 Dactual 5 3.05 1 6.1 1 5.33 5 14.48 m Mmax 5 344.9 kN ? m>m. Now the following table can be prepared. Section PZ-22 PZ-27 4 l (m /m) r 5 10.91 3 Hr4 1027 a b H⬘(m) El 115.2 ⫻ 10⫺6 14.48 251.5 ⫻ 10⫺6 14.48 20.11 ⫻ 10⫺4 9.21 ⫻ 10⫺4 all Md ⫽ S log 3 S(m /m) ⫺2.7 97 ⫻ 10⫺5 ⫺3.04 162.3 ⫻ 10⫺5 (kN ⭈ m /m) Md Mmax 167.33 284.84 0.485 0.826 Figure 9.25 gives a plot of Md/Mmax versus . It can be seen that PZ-27 will be sufficient. 472 Chapter 9: Sheet Pile Walls 1.0 PZ-27 0.8 Md Mmax 0.6 Loose sand PZ-22 0.4 0.2 0 –4.0 –3.5 –3.0 Log –2.5 –2.0 Figure 9.25 Plot of Md/Mmax versus log 9.12 ■ Computational Pressure Diagram Method for Penetration into Sandy Soil The computational pressure diagram (CPD) method for sheet pile penetrating a sandy soil is a simplified method of design and an alternative to the free earth method described in Sections 9.9 and 9.11 (Nataraj and Hoadley, 1984). In this method, the net pressure diagram shown in Figure 9.17 is replaced by rectangular pressure diagrams, as in Figure 9.26. Note that sar is the width of the net active pressure diagram above the dredge line and spr is the width of the net passive pressure diagram below the dredge line. The magnitudes of sar and spr may respectively be expressed as sar 5 CKagav rL (9.75) spr 5 RCKagav r L 5 Rsar (9.76) and where gav r 5 average effective unit weight of sand gL1 1 grL2 < L1 1 L2 C 5 coefficient L(L 2 2l1 ) R 5 coefficient 5 D(2L 1 D 2 2l1 ) The range of values for C and R is given in Table 9.2. (9.77) (9.78) 473 9.12 Computational Pressure Diagram Method for Penetration into Sandy Soil l1 L1 Water table l2 Sand; ␥, F Anchor tie rod Sand L2 ␥sat a Sand D ␥sat p Figure 9.26 Computational pressure diagram method (Note: L1 1 L2 5 L) Table 9.2 Range of Values for C and R [from Eqs. (9.75) and (9.76)] Ca Soil type Loose sand Medium sand Dense sand R 0.8– 0.85 0.7– 0.75 0.55–0.65 0.3–0.5 0.55–0.65 0.60–0.75 a Valid for the case in which there is no surcharge above the granular backfill (i.e., on the right side of the wall, as shown in Figure 9.26) The depth of penetration, D, anchor force per unit length of the wall, F, and maximum moment in the wall, Mmax , are obtained from the following relationships. Depth of Penetration For the depth of penetration, we have D2 1 2DL B 1 2 ¢ l1 l1 L2 ≤ R 2 ¢ ≤ B1 2 2¢ ≤ R 5 0 L R L (9.79) Anchor Force The anchor force is F 5 sar (L 2 RD) (9.80) 474 Chapter 9: Sheet Pile Walls Maximum Moment The maximum moment is calculated from Mmax 5 0.5 sar L2 B ¢1 2 2l1 RD 2 RD ≤ 2 ¢ ≤ ¢1 2 ≤R L L L (9.81) Note the following qualifications: 1. The magnitude of D obtained from Eq. (9.79) is about 1.25 to 1.5 times the value of Dtheory obtained by the conventional free earth support method (see Section 9.9), so D < Dactual c Eq. (9.79) c Eq. (9.68) 2. The magnitude of F obtained by using Eq. (9.80) is about 1.2 to 1.6 times the value obtained by using Eq. (9.66). Thus, an additional factor of safety for the actual design of anchors need not be used. 3. The magnitude of Mmax obtained from Eq. (9.81) is about 0.6 to 0.75 times the value of Mmax obtained by the conventional free earth support method. Hence, the former value of Mmax can be used as the actual design value, and Rowe’s moment reduction need not be applied. Example 9.8 For the anchored sheet pile wall shown in Figure 9.27, determine (a) D, (b) F, and (c) Mmax. Use the CPD method; assume that C 5 0.68 and R 5 0.6. Solution Part a gr 5 gsat 2 gw 5 19.24 2 9.81 5 9.43 kN>m3 From Eq. (9.77) r 5 gav gL1 1 grL2 (17.3) (3) 1 (9.43) (6) 5 5 12.05 kN>m3 L1 1 L2 316 Ka 5 tan2 ¢45 2 fr 35 ≤ 5 tan2 ¢ 45 2 ≤ 5 0.271 2 2 sar 5 CKagav r L 5 (0.68) (0.271) (12.05) (9) 5 19.99 kN>m2 spr 5 Rsar 5 (0.6) (19.99) 5 11.99 kN>m2 From Eq. (9.80) D2 1 2DL B1 2 ¢ l1 l1 L2 ≤R 2 B1 2 2¢ ≤ R 5 0 L R L 9.12 Computational Pressure Diagram Method for Penetration into Sandy Soil L1 3 m Water table l1 1.5 m Anchor 475 Sand c 0 ␥ 17.3 kN/m3 35° L2 6 m Sand ␥sat 19.24 kN/m3 c 0 35° D Sand ␥sat 19.24 kN/m3 c 0 35° Figure 9.27 or D2 1 2(D) (9) B1 2 ¢ (9) 2 1.5 1.5 ≤R 2 B1 2 2¢ ≤ R 5 D2 1 50D 2 1000 5 0 9 0.6 9 Hence D < 4.6 m. Check for the assumption of R: R5 939 2 (2) (1.5)4 L(L 2 2l1 ) 5 < 0.6 —OK D(2L 1 D 2 2l1 ) 4.63(2) (9) 1 4.6 2 (2) (1.5)4 Part b From Eq. (9.80) F 5 sar (L 2 RD) 5 19.9939 2 (0.6) (4.6)4 5 124.74 kN , m Part c From Eq. (9.81) Mmax 5 0.5sar L2 B ¢1 2 12 2l1 RD 2 RD ≤ 2 ¢ ≤ ¢1 2 ≤R L L L (0.6) (4.6) RD 512 5 0.693 L 9 So, Mmax 5 (0.5) (19.99) (9) 2 B (0.693) 2 2 (2) (1.5) (0.693) R 5 201.6 kN-m , m 9 ■ 476 Chapter 9: Sheet Pile Walls 9.13 Fixed Earth-Support Method for Penetration into Sandy Soil When using the fixed earth support method, we assume that the toe of the pile is restrained from rotating, as shown in Figure 9.28a. In the fixed earth support solution, a simplified method called the equivalent beam solution is generally used to calculate L3 and, thus, D. The development of the equivalent beam method is generally attributed to Blum (1931). In order to understand this method, compare the sheet pile to a loaded cantilever beam RSTU, as shown in Figure 9.29. Note that the support at T for the beam is equivalent to the anchor load reaction (F) on the sheet pile (Figure 9.28). It can be seen that the point S of the beam RSTU is the inflection point of the elastic line of the beam, which is equivalent to point I in Figure 9.28. It the beam is cut at S and a free support (reaction Ps) is provided at that point, the bending moment diagram for portion STU of the beam will remain unchanged. This beam STU will be equivalent to the section STU of the beam RSTU. The force P⬘ shown in Figure 9.28a at I will be equivalent to the reaction Ps on the beam (Figure 9.29). The following is an approximate procedure for the design of an anchored sheet-pile wall (Cornfield, 1975). Refer to Figure 9.28. Step 1. Determine L5, which is a function of the soil friction angle ⬘ below the dredge line, from the following: ⴕ (deg) L5 L1 1 L2 30 35 40 0.08 0.03 0 A l1 L1 O′ 1 Deflected shape of sheet pile F Anchor l 2 C Water table Sand γ, φ′ z Sand γsat φ′ L2 2 L5 L3 D L5 I J P′ E D H F B (a) Pressure diagram G (b) Moment diagram FIGURE 9.28 Fixed earth support method for penetration of sandy soil 9.13 Fixed Earth-Support Method for Penetration into Sandy Soil R S T 477 U Beam Ps Moment diagram Figure 9.29 Equivalent cantilever beam concept Step 4. Step 5. Calculate the span of the equivalent beam as l2 ⫹ L2 ⫹ L5 ⫽ L⬘. Calculate the total load of the span, W. This is the area of the pressure diagram between O⬘ and I. Calculate the maximum moment, Mmax, as WL⬘/8. Calculate P⬘ by taking the moment about O⬘, or Step 6. 1 (moment of area ACDJI about O r ) Lr Calculate D as Step 2. Step 3. Pr 5 6P r Å (Kp 2 Ka )gr D 5 L5 1 1.2 Step 7. (9.82) (9.83) Calculate the anchor force per unit length, F, by taking the moment about I, or 1 (moment of area ACDJI about I) F5 Lr Example 9.9 Consider the anchored sheet-pile structure described in Example 9.5. Using the equivalent beam method described in Section 9.13, determine a. Maximum moment b. Theoretical depth of penetration c. Anchor force per unit length of the structure Solution Part a Determination of L5: For ⬘ ⫽ 30°, L5 5 0.08 L1 1 L2 478 Chapter 9: Sheet Pile Walls L5 5 0.08 3.05 1 6.1 L5 ⫽ 0.73 Net Pressure Diagram: From Example 9.5, Ka 5 31 , Kp ⫽ 3, ␥ ⫽ 16 kN/m3, ␥⬘ ⫽ 9.69 kN/m3, 1⬘ ⫽ 16.27 kN/m2, 2⬘ ⫽ 35.97 kN/m2. The net active pressure at a depth L5 below the dredge line can be calculated as 2⬘ ⫺ ␥⬘(Kp ⫺ Ka)L5 ⫽ 35.97 ⫺ (9.69)(3 ⫺ 0.333)(0.73) ⫽ 17.1 kN/m2 The net pressure diagram from z ⫽ 0 to z ⫽ L1 ⫹ L2 ⫹ L5 is shown in Figure 9.30. Maximum Moment: 1 1 W 5 a b (8.16 1 16.27) (1.52) 1 a b (6.1) (16.27 1 35.97) 2 2 1 1 a b (0.73) (35.97 1 17.1) 2 ⫽ 197.2 kN/m L⬘ ⫽ l2 ⫹ L2 ⫹ L5 ⫽ 1.52 ⫹ 6.1 ⫹ 0.73 ⫽ 8.35 m Mmax 5 (197.2) (8.35) WLr 5 5 205.8 kN ? m>m 8 8 Part b Pr 5 1 (moment of area ACDJI about O⬘) Lr A 8.16 kN/m2 O′ 1.53 m = l1 F l2 = 1.52 m 16.27 kN/m2 C 6.1 m = L2 35.97 kN/m2 D L5 = 0.73 m P′ I 17.1 kN/m2 J FIGURE 9.30 9.14 Field Observations for Anchor Sheet Pile Walls 479 - £1≥ 2 6.1 ≥ (16.27) (3.05) £ 3 3.05 2 1.53≥ 1 (16.27) (6.1) £1.52 1 2 3 2 1 1 2 1 P9 5 H1 £ ≥ (6.1) (35.97 2 16.27) £1.52 1 3 6.1≥ 1 £ ≥ (35.97 1 17.1) X 8.35 2 3 2 0.73 ≥ 3 (0.73) £1.52 1 6.1 1 2 c Approximate ⫽ 114.48 kN/m From Eq. (9.83) D 5 L5 1 1.2 (6) (114.48) 6P r 5 0.73 1 1.2 5 6.92 m Å (Kp 2 Ka )gr Å (3 2 0.333) (9.69) Part c Taking the moment about I (Figure 9.30) 1 3.05 6.1 a (16.27) (3.05) a0.73 1 6.1 1 b 1 (16.27) (6.1) a0.73 1 b 1 2 3 2 F5 E U 8.35 1 6.1 1 0.73 1 a b (6.1) (35.97 2 16.27) a0.73 1 b 1 a b (35.97 1 17.1) (0.73) a b 2 3 2 2 c Approximate ⫽ 88.95 kN/m 9.14 ■ Field Observations for Anchor Sheet Pile Walls In the preceding sections, large factors of safety were used for the depth of penetration, D. In most cases, designers use smaller magnitudes of soil friction angle, ⬘, thereby ensuring a built-in factor of safety for the active earth pressure. This procedure is followed primarily because of the uncertainties involved in predicting the actual earth pressure to which a sheet-pile wall in the field will be subjected. In addition, Casagrande (1973) observed that, if the soil behind the sheet-pile wall has grain sizes that are predominantly smaller than those of coarse sand, the active earth pressure after construction sometimes increases to an at-rest earth-pressure condition. Such an increase causes a large increase in the anchor force, F. The following two case histories are given by Casagrande (1973). Bulkhead of Pier C—Long Beach Harbor, California (1949) A typical cross section of the Pier C bulkhead of the Long Beach harbor is shown in Figure 9.31. Except for a rockfill dike constructed with 76 mm (3 in.) maximum-size quarry wastes, the backfill of the sheet-pile wall consisted of fine sand. Figure 9.32 shows the 480 Chapter 9: Sheet Pile Walls +5.18 m +1.22 m Tie rod –76 mm dia. Fine sand –3.05 m hydraulic fill El.0 Mean low water level MZ 38 Steel sheet pile 1V: 1.5 H 1V: 0.58H Rock dike 76 mm maximum size –11.56 m Fine sand –18.29 m 0 10 m Scale Figure 9.31 Pier C bulkhead—Long Beach harbor (Adapted after Casagrande, 1973) variation of the lateral earth pressure between May 24, 1949 (the day construction was completed) and August 6, 1949. On May 24, the lateral earth pressure reached an active state, as shown in Figure 9.32a, due to the wall yielding. Between May 24 and June 3, the anchor resisted further yielding and the lateral earth pressure increased to the at-rest state (Figure 9.32b). However, the flexibility of the sheet piles ultimately resulted in a gradual decrease in the lateral earth-pressure distribution on the sheet piles (see Figure 9.32c). May 24 June 3 August 6 +5.18 m 151.11 MN/m2 +5.18 m 177.33 MN/m2 +5.18 m 213.9 MN/m2 –3.05 m –3.05 m –11.56 m –11.56 m (a) 0 –3.05 m (b) 100 –11.56 m (c) 200 kN/m2 Pressure scale Figure 9.32 Measured stresses at Station 27 ⫹ 30—Pier C bulkhead, Long Beach (Adapted after Casagrande, 1973) 9.14 Field Observations for Anchor Sheet Pile Walls 481 With time, the stress on the tie rods for the anchor increased as shown in the following table. Stress on anchor tie rod (MN/m2) Date May 24, 1949 June 3, 1949 June 11, 1949 July 12, 1949 August 6, 1949 151.11 177.33 193.2 203.55 213.9 These observations show that the magnitude of the active earth pressure may vary with time and depend greatly on the flexibility of the sheet piles. Also, the actual variations in the lateral earth-pressure diagram may not be identical to those used for design. Bulkhead—Toledo, Ohio (1961) A typical cross section of a Toledo bulkhead completed in 1961 is shown in Figure 9.33. The foundation soil was primarily fine to medium sand, but the dredge line did cut into highly overconsolidated clay. Figure 9.33 also shows the actual measured values of bending moment along the sheet-pile wall. Casagrande (1973) used the Rankine active earthpressure distribution to calculate the maximum bending moment according to the free earth support method with and without Rowe’s moment reduction. Maximum predicted bending moment, Mmax Design method Free earth support method Free earth support method with Rowe’s moment reduction Top of fill 0 2 146.5 kN-m 78.6 kN-m Scale 0 Tie rod 100 200 kN-m 81 kN-m 4 65 kN-m May 1961 6 180 kN-m 8 10 205 kN-m 12 14 Depth (m) Dredge line Figure 9.33 Bending moment from straingage measurements at test location 3, Toledo bulkhead (Adapted after Casagrande, 1973) 482 Chapter 9: Sheet Pile Walls Comparisons of these magnitudes of Mmax with those actually observed show that the field values are substantially larger. The reason probably is that the backfill was primarily fine sand and the measured active earth-pressure distribution was larger than that predicted theoretically. 9.15 Free Earth Support Method for Penetration of Clay Figure 9.34 shows an anchored sheet-pile wall penetrating a clay soil and with a granular soil backfill. The diagram of pressure distribution above the dredge line is similar to that shown in Figure 9.12. From Eq. (9.42), the net pressure distribution below the dredge line (from z 5 L1 1 L2 to z 5 L1 1 L2 1 D) is s6 5 4c 2 (gL1 1 grL2 ) For static equilibrium, the sum of the forces in the horizontal direction is P1 2 s6D 5 F (9.84) where P1 5 area of the pressure diagram ACD F 5 anchor force per unit length of the sheet pile wall A l1 L1 O F 1 Water level l2 C Sand, ␥, z Sand ␥sat, L2 P1 z1 2 Dredge line D E Clay Clay ␥sat =0 c D F 6 B Figure 9.34 Anchored sheet-pile wall penetrating clay 9.15 Free Earth Support Method for Penetration of Clay 483 Again, taking the moment about Or produces P1 (L1 1 L2 2 l1 2 z1 ) 2 s6D¢l2 1 L2 1 D ≤ 50 2 Simplification yields s6D2 1 2s6D(L1 1 L2 2 l1 ) 2 2P1 (L1 1 L2 2 l1 2 z1 ) 5 0 (9.85) Equation (9.85) gives the theoretical depth of penetration, D. As in Section 9.9, the maximum moment in this case occurs at a depth L1 , z , L1 1 L2 . The depth of zero shear (and thus the maximum moment) may be determined from Eq. (9.69). A moment reduction technique similar to that in Section 9.11 for anchored sheet piles penetrating into clay has also been developed by Rowe (1952, 1957). This technique is presented in Figure 9.35, in which the following notation is used: 1. The stability number is Sn 5 1.25 c (gL1 1 grL2 ) (9.86) where c 5 undrained cohesion (f 5 0) . For the definition of g, gr, L1 , and L2 , see Figure 9.34. 2. The nondimensional wall height is a5 L1 1 L2 L1 1 L2 1 Dactual (9.87) 3. The flexibility number is r [see Eq. (9.74)] 4. Md 5 design moment Mmax 5 maximum theoretical moment The procedure for moment reduction, using Figure 9.35, is as follows: Obtain Hr 5 L1 1 L2 1 Dactual . Determine a 5 (L1 1 L2 )>Hr. Determine Sn [from Eq. (9.86)]. For the magnitudes of a and Sn obtained in Steps 2 and 3, determine Md>Mmax for various values of log r from Figure 9.35, and plot Md>Mmax against log r. Step 5. Follow Steps 1 through 9 as outlined for the case of moment reduction of sheet-pile walls penetrating granular soil. (See Section 9.11.) Step 1. Step 2. Step 3. Step 4. 484 Chapter 9: Sheet Pile Walls 1.0 Log = –3.1 0.8 Md Mmax ␣ = 0.8 0.7 0.6 0.6 0.4 1.0 Log = –2.6 0.8 Md Mmax ␣ = 0.8 0.6 0.6 0.7 0.4 1.0 Log = –2.0 0.8 Figure 9.35 Plot of Md>Mmax against stability number for sheet-pile wall penetrating clay (From Rowe, P. W. (1957). “Sheet Pile Walls in Clay,” Proceedings, Institute of Civil Engineers, Vol. 7, pp. 654–692.) Md Mmax ␣ = 0.8 0.6 0.7 0.6 0.4 0 0.5 1.0 Stability number, Sn 1.5 1.75 Example 9.10 In Figure 9.34, let L1 5 3 m, L2 5 6 m, and l1 5 1.5 m. Also, let g 5 17 kN>m3, gsat 5 20 kN>m3, fr 5 35°, and c 5 41 kN>m2. a. Determine the theoretical depth of embedment of the sheet-pile wall. b. Calculate the anchor force per unit length of the wall. Solution Part a We have Ka 5 tan2 ¢ 45 2 fr 35 ≤ 5 tan2 ¢45 2 ≤ 5 0.271 2 2 9.15 Free Earth Support Method for Penetration of Clay 485 and Kp 5 tan2 ¢ 45 1 fr 35 ≤ 5 tan2 ¢ 45 1 ≤ 5 3.69 2 2 From the pressure diagram in Figure 9.36, s1r 5 gL1Ka 5 (17) (3) (0.271) 5 13.82 kN>m2 s2r 5 (gL1 1 grL2 )Ka 5 3(17) (3) 1 (20 2 9.81) (6) 4 (0.271) 5 30.39 kN>m2 P1 5 areas 1 1 2 1 3 5 1>2(3) (13.82) 1 (13.82) (6) 1 1>2(30.39 2 13.82) (6) 5 20.73 1 82.92 1 49.71 5 153.36 kN>m and 3 6 6 (20.73) ¢ 6 1 ≤ 1 (82.92) ¢ ≤ 1 (49.71) ¢ ≤ 3 2 3 z1 5 5 3.2 m 153.36 From Eq. (9.85), s6D2 1 2s6D(L1 1 L2 2 l1 ) 2 2P1 (L1 1 L2 2 l1 2 z1 ) 5 0 s6 5 4c 2 (gL1 1 grL2 ) 5 (4) (41) 2 3(17) (3) 1 (20 2 9.81) (6)4 5 51.86 kN>m2 So, (51.86)D2 1 (2) (51.86) (D) (3 1 6 2 1.5) 2 (2) (153.36) (3 1 6 2 1.5 2 3.2) 5 0 l1 1.5 m L1 3 m l2 1.5 m 1 1 13.82 kN/m2 L2 6 m 2 3 2 30.39 kN/m2 1.6 m D 6 51.86 kN/m2 Figure 9.36 Free earth support method, sheet pile penetrating into clay 486 Chapter 9: Sheet Pile Walls or D2 1 15D 2 25.43 5 0 Hence, D < 1.6 m Part b From Eq. (9.84), F 5 P1 2 s6D 5 153.36 2 (51.86) (1.6) 5 70.38 kN , m 9.16 ■ Anchors Sections 9.9 through 9.15 gave an analysis of anchored sheet-pile walls and discussed how to obtain the force F per unit length of the sheet-pile wall that has to be sustained by the anchors. The current section covers in more detail the various types of anchor generally used and the procedures for evaluating their ultimate holding capacities. The general types of anchor used in sheet-pile walls are as follows: 1. 2. 3. 4. Anchor plates and beams (deadman) Tie backs Vertical anchor piles Anchor beams supported by batter (compression and tension) piles Anchor plates and beams are generally made of cast concrete blocks. (See Figure 9.37a.) The anchors are attached to the sheet pile by tie-rods. A wale is placed at the front or back face of a sheet pile for the purpose of conveniently attaching the tie-rod to the wall. To protect the tie rod from corrosion, it is generally coated with paint or asphaltic materials. In the construction of tiebacks, bars or cables are placed in predrilled holes (see Figure 9.37b) with concrete grout (cables are commonly high-strength, prestressed steel tendons). Figures 9.37c and 9.37d show a vertical anchor pile and an anchor beam with batter piles. Placement of Anchors The resistance offered by anchor plates and beams is derived primarily from the passive force of the soil located in front of them. Figure 9.37a, in which AB is the sheet-pile wall, shows the best location for maximum efficiency of an anchor plate. If the anchor is placed inside wedge ABC, which is the Rankine active zone, it would not provide any resistance to failure. Alternatively, the anchor could be placed in zone CFEH. Note that line DFG is the slip line for the Rankine passive pressure. If part of the passive wedge is located inside the active wedge ABC, full passive resistance of the anchor cannot be realized upon 9.16 Anchors 45 /2 A 45 /2 D Groundwater table 45 /2 I Anchor F plate G H or beam 487 Sheet pile C Anchor plate or beam E Wale Tie rod B Section Plan (a) 45 /2 45 /2 45 /2 Tie rod Groundwater table Groundwater table Tie rod or cable Anchor pile Concrete grout (b) (c) 45 /2 Groundwater table Tie rod Anchor beam Compression pile Tension pile (d) Figure 9.37 Various types of anchoring for sheet-pile walls: (a) anchor plate or beam; (b) tieback; (c) vertical anchor pile; (d) anchor beam with batter piles failure of the sheet-pile wall. However, if the anchor is placed in zone ICH, the Rankine passive zone in front of the anchor slab or plate is located completely outside the Rankine active zone ABC. In this case, full passive resistance from the anchor can be realized. Figures 9.37b, 9.37c, and 9.37d also show the proper locations for the placement of tiebacks, vertical anchor piles, and anchor beams supported by batter piles. 488 Chapter 9: Sheet Pile Walls 9.17 Holding Capacity of Anchor Plates in Sand Semi-Empirical Method Ovesen and Stromann (1972) proposed a semi-empirical method for determining the ultimate resistance of anchors in sand. Their calculations, made in three steps, are carried out as follows: Step 1. Basic Case. Determine the depth of embedment, H. Assume that the anchor slab has height H and is continuous (i.e., B 5 length of anchor slab perpendicular to the cross section 5 ` ), as shown in Figure 9.38, in which the following notation is used: Pp 5 passive force per unit length of anchor Pa 5 active force per unit length of anchor fr 5 effective soil friction angle dr 5 friction angle between anchor slab and soil Pult r 5 ultimate resistance per unit length of anchor W 5 effective weight per unit length of anchor slab Also, r 5 12 gH 2Kp cos d r 2 Pa cos fr 5 12 gH 2Kp cos dr 2 12 gH 2Ka cos fr Pult 5 12 gH 2 (Kp cos dr 2 Ka cos fr) (9.88) where Ka 5 active pressure coefficient with dr 5 fr (see Figure 9.39a) Kp 5 passive pressure coefficient To obtain Kp cos dr, first calculate Kp sin dr 5 W 1 Pa sin fr 1 2 2 gH 5 W 1 12gH 2Ka sin fr 45 /2 Pa H (9.89) 1 2 2 gH 45 /2 Pult Sand ␦ Pp ␥ Figure 9.38 Basic case: continuous vertical anchor in granular soil 9.17 Holding Capacity of Anchor Plates in Sand 489 0.7 0.6 0.5 Pa 0.4 Arc of log spiral Ka 0.3 0.2 0.1 10 40 20 30 Soil friction angle, (deg) (a) 45 14 12 45 10 40 8 Kp cos ␦ 35 6 30 4 = 25 3 2 0 1 2 3 4 5 Kp sin ␦ (b) Figure 9.39 (a) Variation of Ka for dr 5 fr, (b) variation of Kp cos dr with Kp sin d r (Based on Ovesen and Stromann, 1972) Then use the magnitude of Kp sin dr obtained from Eq. (9.89) to estimate the magnitude of Kp cos dr from the plots given in Figure 9.39b. Step 2. Strip Case. Determine the actual height h of the anchor to be constructed. If a continuous anchor (i.e., an anchor for which B 5 ` ) of height h is placed in the soil so that its depth of embedment is H, as shown in Figure 9.40, the ultimate resistance per unit length is 490 Chapter 9: Sheet Pile Walls Sand ␥ H h Pus Figure 9.40 Strip case: vertical anchor Pus r 5 C Cov 1 1 H Cov 1 ¢ ≤ h S Pult r c (9.90) Eq. 9.88 where P rus 5 ultimate resistance for the strip case Cov 5 19 for dense sand and 14 for loose sand Step 3. Actual Case. In practice, the anchor plates are placed in a row with centerto-center spacing Sr, as shown in Figure 9.41a. The ultimate resistance of each anchor is Sand ␥ B H S S h (a) 0.5 Dense sand (Be – B)/(H + h) 0.4 0.3 Loose sand 0.2 0.1 0 0 0.5 (S – B)/(H – h) (b) 1.0 1.25 Figure 9.41 (a) Actual case for row of anchors; (b) variation of (Be 2 B)> (H 1 h) with (Sr 2 B)> (H 1 h) (Based on Ovesen and Stromann, 1972) 9.17 Holding Capacity of Anchor Plates in Sand Pult 5 Pus r Be 491 (9.91) where Be 5 equivalent length. The equivalent length is a function of Sr, B, H, and h. Figure 9.41b shows a plot of (Be 2 B)> (H 1 h) against (Sr 2 B)> (H 1 h) for the cases of loose and dense sand. With known values of Sr, B, H, and h, the value of Be can be calculated and used in Eq. (9.91) to obtain Pult . Stress Characteristic Solution Neely, Stuart, and Graham (1973) proposed a stress characteristic solution for anchor pullout resistance using the equivalent free surface concept. Figure 9.42 shows the assumed failure surface for a strip anchor. In this figure, OX is the equivalent free surface. The shear stress (so) mobilized along OX can be given as m5 so 9 so tan f9 (9.92) where m ⫽ shear stress mobilization factor o⬘ ⫽ effective normal stress along OX Using this analysis, the ultimate resistance (Pult) of an anchor (length ⫽ B and height ⫽ h) can be given as Pult ⫽ M␥q (␥h2)BFs (9.93) where M␥q ⫽ force coefficient Fs ⫽ shape factor ␥ ⫽ effective unit weight of soil The variations of M␥q for m ⫽ 0 and 1 are shown in Figure 9.43. For conservative design, M␥q with m ⫽ 0 may be used. The shape factor (Fs) determined experimentally is shown in Figure 9.44 as a function of B/h and H/h. X o so ␥ O H Pult h m= so o tan Figure 9.42 Assumed failure surface in soil for stress characteristic solution 492 Chapter 9: Sheet Pile Walls 200 100 M␥q (log scale) 50 20 = 45° 10 = 40° = 35° 5 m=0 m=1 = 30° 2 1 0 1 2 4 3 5 H/h Figure 9.43 Variation of M␥q with H/h and ⬘ (After Neeley et al., 1973. With permission from ASCE.) 2.5 B/h = 1.0 Shape factor, Fs 2.0 1.5 2.0 2.75 3.5 5.0 1.0 0.5 0 1 2 3 H/h 4 5 Figure 9.44 Variation of shape factor with H/h and B/h (After Neeley et al., 1973. With permission from ASCE.) Empirical Correlation Based on Model Tests Ghaly (1997) used the results of 104 laboratory tests, 15 centrifugal model tests, and 9 field tests to propose an empirical correlation for the ultimate resistance of single anchors. The correlation can be written as Pult 5 5.4 H 2 0.28 ¢ ≤ gAH tan fr A where A 5 area of the anchor 5 Bh. (9.94) 9.17 Holding Capacity of Anchor Plates in Sand 493 Ghaly also used the model test results of Das and Seeley (1975) to develop a load–displacement relationship for single anchors. The relationship can be given as P u 0.3 5 2.2¢ ≤ Pult H (9.95) where u 5 horizontal displacement of the anchor at a load level P. Equations (9.94) and (9.95) apply to single anchors (i.e., anchors for which Sr>B 5 `). For all practical purposes, when Sr>B < 2 the anchors behave as single anchors. Factor of Safety for Anchor Plates The allowable resistance per anchor plate may be given as Pall 5 Pult FS where FS 5 factor of safety. Generally, a factor of safety of 2 is suggested when the method of Ovesen and Stromann is used. A factor of safety of 3 is suggested for Pult calculated by Eq. (9.94). Spacing of Anchor Plates The center-to-center spacing of anchors, Sr, may be obtained from Sr 5 Pall F where F 5 force per unit length of the sheet pile. Example 9.11 Refer to Figure 9.41a. Given: B ⫽ h ⫽ 0.4 m, S⬘ ⫽ 1.2 m, H ⫽ 1 m, ␥ ⫽ 16.51 kN/m3, and ⬘ ⫽ 35°. Determine the ultimate resistance for each anchor plate. The anchor plates are made of concrete and have thicknesses of 0.15 m. Solution From Figure 9.39a for ⬘ ⫽ 35°, the magnitude of Ka is about 0.26. W ⫽ Ht␥concrete ⫽ (1 m)(0.15 m)(23.5 kN/m3) ⫽ 3.525 kN/m From Eq. (9.89), Kp sin d9 5 5 W 1 1> 2gH 2Ka sin f9 > 2gH 2 3.525 1 (0.5) (16.51) (1) 2 (0.26) (sin 35) 1 (0.5) (16.51) (1) 2 5 0.576 494 Chapter 9: Sheet Pile Walls From Figure 9.39b with ⬘ ⫽ 35° and Kp sin ␦⬘ ⫽ 0.576, the value of Kp cos ␦⬘ is about 4.5. Now, using Eq. (9.88), Pult ⬘ ⫽ 21␥H2(Kp cos ␦⬘ ⫺ Ka cos ⬘) ⫽ (12)(16.51)(1)2[4.5 ⫺ (0.26)(cos 35)] ⫽ 35.39 kN/m In order to calculate Pus ⬘, let us assume the sand to be loose. So, Cov in Eq. (9.90) is equal to 14. Hence, Pus r 5D Cov 1 1 H Cov 1 a b h T Pult r 5D 14 1 1 T 5 32.17 kN>m 1 14 1 a b 0.4 Sr 2 B 1.2 2 0.4 0.8 5 5 5 0.571 H1h 1 1 0.4 1.4 For (S⬘ ⫺ B)/(H ⫹ h) ⫽ 0.571 and loose sand, Figure 9.41b yields Be 2 B 5 0.229 H2h So Be ⫽ (0.229)(H ⫹ h) ⫹ B ⫽ (0.229)(1 ⫹ 0.4) ⫹ 0.4 ⫽ 0.72 Hence, from Eq. (9.91) Pult ⫽ Pus ⬘ Be ⫽ (32.17)(0.72) ⫽ 23.16 kN ■ Example 9.12 Refer to a single anchor given in Example 9.11 using the stress characteristic solution. Estimate the ultimate anchor resistance. Use m ⫽ 0 in Figure 9.43. Solution Given: B ⫽ h ⫽ 0.4 m and H ⫽ 1 m. Thus, H 1m 5 5 2.5 h 0.4 m 0.4 m B 5 51 h 0.4 m 9.19 Ultimate Resistance of Tiebacks 495 From Eq. (9.93), Pult ⫽ M␥q␥h2 BFs From Figure 9.43, with ⬘ ⫽ 35° and H/h ⫽ 2.5, M␥q ⬇ 18.2. Also, from Figure 9.44, with H/h ⫽ 2.5 and B/h ⫽ 1, Fs ⬇ 1.8. Hence, Pult ⫽ (18.2)(16.51)(0.4)2(0.4)(1.8) ⬇ 34.62 kN ■ Example 9.13 Solve Example Problem 9.12 using Eq. (9.94). Solution From Eq. (9.94), Pult 5 5.4 H 2 0.28 a b gAH tan f9 A H⫽1m A ⫽ Bh ⫽ (0.4 ⫻ 0.4) ⫽ 0.16 m2 Pult 5 9.18 5.4 (1) 2 0.28 c d (16.51) (0.16) (1) < 34.03 kN tan 35 0.16 ■ Holding Capacity of Anchor Plates in Clay (f 5 0 Condition) Relatively few studies have been conducted on the ultimate resistance of anchor plates in clayey soils (f 5 0). Mackenzie (1955) and Tschebotarioff (1973) identified the nature of variation of the ultimate resistance of strip anchors and beams as a function of H, h, and c (undrained cohesion based on f 5 0) in a nondimensional form based on laboratory model test results. This is shown in the form of a nondimensional plot in Figure 9.45 (Pult>hBc versus H>h) and can be used to estimate the ultimate resistance of anchor plates in saturated clay (f 5 0). 9.19 Ultimate Resistance of Tiebacks According to Figure 9.46, the ultimate resistance offered by a tieback in sand is Pult 5 pdl sor K tan fr (9.96) Chapter 9: Sheet Pile Walls 12 10 8 Pult hBc 496 6 4 2 0 0 5 10 15 20 H h Pult with H>h for plate anchors in clay hBc (Based on Mackenzie (1955) and Tschebotarioff (1973)) Figure 9.45 Experimental variation of z l d Figure 9.46 Parameters for defining the ultimate resistance of tiebacks where fr 5 effective angle of friction of soil sor 5 average effective vertical stress (5gz in dry sand) K 5 earth pressure coefficient The magnitude of K can be taken to be equal to the earth pressure coefficient at rest (Ko ) if the concrete grout is placed under pressure (Littlejohn, 1970). The lower limit of K can be taken to be equal to the Rankine active earth pressure coefficient. In clays, the ultimate resistance of tiebacks may be approximated as Pult 5 pdlca where ca 5 adhesion. (9.97) Problems 497 The value of ca may be approximated as 23cu (where cu 5 undrained cohesion). A factor of safety of 1.5 to 2 may be used over the ultimate resistance to obtain the allowable resistance offered by each tieback. Problems 9.1 9.2 9.3 9.4 9.5 9.6 Figure P9.1 shows a cantilever sheet pile wall penetrating a granular soil. Here, L1 5 4 m, L2 5 8 m, g 5 16.1 kN>m3, gsat 5 18.2 kN>m3, and f r 5 32°. a. What is the theoretical depth of embedment, D? b. For a 30% increase in D, what should be the total length of the sheet piles? c. Determine the theoretical maximum moment of the sheet pile. Redo Problem 9.1 with the following: L1 5 3 m, L2 5 6 m, g 5 17.3 kN>m3, gsat 5 19.4 kN>m3 , and fr 5 30°. Refer to Figure 9.10. Given: L 5 3 m, g 5 16.7 kN>m3, and fr 5 30°. Calculate the theoretical depth of penetration, D, and the maximum moment. Refer to Figure P9.4, for which L1 5 2.4 m, L2 5 4.6 m, g 5 15.7 kN>m3 , gsat 5 17.3 kN>m3 , and f r 5 30°, and c 5 29 kN>m2. a. What is the theoretical depth of embedment, D? b. Increase D by 40%. What length of sheet piles is needed? c. Determine the theoretical maximum moment in the sheet pile. Refer to Figure 9.14. Given: L 5 4 m ; for sand, g 5 16 kN>m3; fr 5 35°; and, for clay, gsat 5 19.2 kN>m3 and c 5 45 kN>m2. Determine the theoretical value of D and the maximum moment. An anchored sheet pile bulkhead is shown in Figure P9.6. Let L1 5 4 m, L2 5 9 m, l1 5 2 m , g 5 17 kN>m3, gsat 5 19 kN>m3 , and fr 5 34°. a. Calculate the theoretical value of the depth of embedment, D. b. Draw the pressure distribution diagram. c. Determine the anchor force per unit length of the wall. Use the free earth-support method. L1 Sand ␥ c 0 L2 Sand ␥sat c 0 D Sand ␥sat c 0 Water table Dredge line Figure P9.1 498 Chapter 9: Sheet Pile Walls L1 Sand ␥ c 0 L2 Sand ␥sat c 0 Water table Clay c 0 D Figure P9.4 l1 Water table L1 Anchor Sand c 0 ␥ L2 Sand ␥sat c 0 D Sand ␥sat c 0 Figure P9.6 9.7 9.8 9.9 In Problem 9.6, assume that Dactual 5 1.3Dtheory . a. Determining the theoretical maximum moment. b. Using Rowe’s moment reduction technique, choose a sheet pile section. Take E 5 210 3 103 MN>m2 and sall 5 210,000 kN>m2. Refer to Figure P9.6. Given: L1 5 4 m, L2 5 8 m, l1 5 l2 5 2 m, ␥ 5 16 kN/m3, ␥sat 5 18.5 kN/m3, and fr 5 35°. Use the charts presented in Section 9.10 and determine: a. Theoretical depth of penetration b. Anchor force per unit length c. Maximum moment in the sheet pile. Refer to Figure P9.6, for which L1 5 4 m, L2 5 7 m, l1 5 1.5 m, g 5 18 kN>m3, gsat 5 19.5 kN>m3, and fr 5 30°. Use the computational diagram method (Section 9.12) to determine D, F, and Mmax. Assume that C 5 0.68 and R 5 0.6. Problems 499 9.10 An anchored sheet-pile bulkhead is shown in Figure P9.10. Let L1 5 2 m, L2 5 6 m, l1 5 1 m, g 5 16 kN>m3, gsat 5 18.86 kN>m3, fr 5 32°, and c 5 27 kN>m2. a. Determine the theoretical depth of embedment, D. b. Calculate the anchor force per unit length of the sheet-pile wall. Use the free earth support method. 9.11 In Figure 9.41a, for the anchor slab in sand, H 5 1.52 m, h 5 0.91 m, B 5 1.22 m, S r 5 2.13 m, fr 5 30°, and g 5 17.3 kN>m3. The anchor plates are made of concrete and have a thickness of 76 mm. Using Ovesen and Stromann’s method, calculate the ultimate holding capacity of each anchor. Take gconcrete 5 23.58 kN>m3. 9.12 A single anchor slab is shown in Figure P9.12. Here, H 5 0.9 m, h 5 0.3 m, g 5 17 kN>m3, and fr 5 32°. Calculate the ultimate holding capacity of the anchor slab if the width B is (a) 0.3 m, (b) 0.6 m, and (c) 5 0.9 m. (Note: center-to-center spacing, S r 5 ` .) Use the empirical correlation given in Section 9.17 [Eq. (9.94)]. 9.13 Repeat Problem 9.12 using Eq. (9.93). Use m 5 0 in Figure 9.43. l1 Anchor L1 Water table Sand c 0 ␥ L2 Sand ␥sat c 0 D Clay c 0 Figure P9.10 ␥ c 0 H h Pult Figure P9.12 500 Chapter 9: Sheet Pile Walls References BLUM, H. (1931) Einspannungsverhältnisse bei Bohlwerken, W. Ernst und Sohn, Berlin, Germany. CASAGRANDE, L. (1973). “Comments on Conventional Design of Retaining Structures,” Journal of the Soil Mechanics and Foundations Division, ASCE, Vol. 99, No. SM2, pp. 181–198. CORNFIELD, G. M. (1975). “Sheet Pile Structures,” in Foundation Engineering Handbook, ed. H. F. Wintercorn and H. Y. Fang, Van Nostrand Reinhold, New York, pp. 418–444. DAS, B. M., and SEELEY, G. R. (1975). “Load–Displacement Relationships for Vertical Anchor Plates,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 101, No, GT7, pp. 711–715. GHALY, A. M. (1997). “Load–Displacement Prediction for Horizontally Loaded Vertical Plates.” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 123, No. 1, pp. 74–76. HAGERTY, D. J., and NOFAL, M. M. (1992). “Design Aids: Anchored Bulkheads in Sand,” Canadian Geotechnical Journal, Vol. 29, No. 5, pp. 789–795. LITTLEJOHN, G. S. (1970). “Soil Anchors,” Proceedings, Conference on Ground Engineering, Institute of Civil Engineers, London, pp. 33–44. MACKENZIE, T. R. (1955). Strength of Deadman Anchors in Clay, M.S. Thesis, Princeton University, Princeton, N. J. NATARAJ, M. S., and HOADLEY, P. G. (1984). “Design of Anchored Bulkheads in Sand,” Journal of Geotechnical Engineering, American Society of Civil Engineers, Vol. 110, No. GT4, pp. 505–515. NEELEY, W. J., STUART, J. G., and GRAHAM, J. (1973). “Failure Loads of Vertical Anchor Plates is Sand,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 99, No. SM9, pp. 669–685. OVESEN, N. K., and STROMANN, H. (1972). “Design Methods for Vertical Anchor Slabs in Sand,” Proceedings, Specialty Conference on Performance of Earth and Earth-Supported Structures. American Society of Civil Engineers, Vol. 2.1, pp. 1481–1500. ROWE, P. W. (1952). “Anchored Sheet Pile Walls,” Proceedings, Institute of Civil Engineers, Vol. 1, Part 1, pp. 27–70. ROWE, P. W. (1957). “Sheet Pile Walls in Clay,” Proceedings, Institute of Civil Engineers, Vol. 7, pp. 654 –692. TSCHEBOTARIOFF, G. P. (1973). Foundations, Retaining and Earth Structures, 2nd ed., McGraw-Hill, New York. TSINKER, G. P. (1983). “Anchored Street Pile Bulkheads: Design Practice,” Journal of Geotechnical Engineering, American Society of Civil Engineers, Vol. 109, No. GT8, pp. 1021–1038. 10 10.1 Braced Cuts Introduction Sometimes construction work requires ground excavations with vertical or near-vertical faces—for example, basements of buildings in developed areas or underground transportation facilities at shallow depths below the ground surface (a cut-and-cover type of construction). The vertical faces of the cuts need to be protected by temporary bracing systems to avoid failure that may be accompanied by considerable settlement or by bearing capacity failure of nearby foundations. Figure 10.1 shows two types of braced cut commonly used in construction work. One type uses the soldier beam (Figure 10.1a), which is driven into the ground before excavation and is a vertical steel or timber beam. Laggings, which are horizontal timber planks, are placed between soldier beams as the excavation proceeds. When the excavation reaches the desired depth, wales and struts (horizontal steel beams) are installed. The struts are compression members. Figure 10.1b shows another type of braced excavation. In this case, interlocking sheet piles are driven into the soil before excavation. Wales and struts are inserted immediately after excavation reaches the appropriate depth. Figure 10.2 shows the braced-cut construction used for the Chicago subway in 1940. Timber lagging, timber struts, and steel wales were used. Figure 10.3 shows a braced cut made during the construction of the Washington, DC, metro in 1974. In this cut, timber lagging, steel H-soldier piles, steel wales, and pipe struts were used. To design braced excavations (i.e., to select wales, struts, sheet piles, and soldier beams), an engineer must estimate the lateral earth pressure to which the braced cuts will be subjected. The theoretical aspects of the lateral earth pressure on a braced cut were discussed in Section 7.8. The total active force per unit length of the wall (Pa ) was calculated using the general wedge theory. However, that analysis does not provide the relationships required for estimating the variation of lateral pressure with depth, which is a function of several factors, such as the type of soil, the experience of the construction crew, the type of construction equipment used, and so forth. For that reason, empirical pressure envelopes developed from field observations are used for the design of braced cuts. This procedure is discussed in the next section. 501 502 Chapter 10: Braced Cuts Wale Strut Strut A A Soldier beam Lagging Lagging Wale Wedge Elevation Plan (a) Wale Strut Strut A A Sheet pile Wale Plan Elevation (b) Figure 10.1 Types of braced cut: (a) use of soldier beams; (b) use of sheet piles 10.2 Pressure Envelope for Braced-Cut Design As mentioned in Section 10.1, the lateral earth pressure in a braced cut is dependent on the type of soil, construction method, and type of equipment used. The lateral earth pressure changes from place to place. Each strut should also be designed for the maximum load to 10.2 Pressure Envelope for Braced-Cut Design 503 Figure 10.2 Braced cut in Chicago Subway construction, January 1940 (Courtesy of Ralph B. Peck) which it may be subjected. Therefore, the braced cuts should be designed using apparentpressure diagrams that are envelopes of all the pressure diagrams determined from measured strut loads in the field. Figure 10.4 shows the method for obtaining the apparent-pressure diagram at a section from strut loads. In this figure, let P1 , P2 , P3 , P4 , c be the measured strut loads. The apparent horizontal pressure can then be calculated as P1 s1 5 d2 (s) ¢ d1 1 ≤ 2 P2 s2 5 d2 d3 (s) ¢ 1 ≤ 2 2 P3 s3 5 d3 d4 (s) ¢ 1 ≤ 2 2 P4 s4 5 d4 d5 (s) ¢ 1 ≤ 2 2 504 Chapter 10: Braced Cuts Image not available due to copyright restrictions d1 d1 1 P1 d2/2 d2 d2/2 2 P2 d3/2 d3 d3/2 P3 3 d4/2 d4 d4/2 P4 d5 d5/2 d5/2 4 Figure 10.4 Procedure for calculating apparent-pressure diagram from measured strut loads 10.2 Pressure Envelope for Braced-Cut Design 505 where s1 , s2 , s3 , s4 5 apparent pressures s 5 center-to-center spacing of the struts Using the procedure just described for strut loads observed from the Berlin subway cut, Munich subway cut, and New York subway cut, Peck (1969) provided the envelope of apparent-lateral-pressure diagrams for design of cuts in sand. This envelope is illustrated in Figure 10.5, in which sa 5 0.65gHKa (10.1) where g 5 unit weight H 5 height of the cut Ka 5 Rankine active pressure coefficient 5 tan2 (45 2 fr>2) fr 5 effective friction angle of sand Cuts in Clay In a similar manner, Peck (1969) also provided the envelopes of apparent-lateral-pressure diagrams for cuts in soft to medium clay and in stiff clay. The pressure envelope for soft to medium clay is shown in Figure 10.6 and is applicable to the condition gH .4 c where c 5 undrained cohesion (f 5 0). The pressure, sa , is the larger of sa 5 gHB1 2 ¢ 4c ≤R gH and sa 5 0.3gH (10.2) where g 5 unit weight of clay. The pressure envelope for cuts in stiff clay is shown in Figure 10.7, in which sa 5 0.2gH to 0.4gH is applicable to the condition gH>c < 4. (with an average of 0.3gH) (10.3) 506 Chapter 10: Braced Cuts 0.25 H 0.25 H a H a 0.5 H a 0.75 H 0.25 H Figure 10.5 Peck’s (1969) apparent-pressure envelope for cuts in sand Figure 10.6 Peck’s (1969) apparent-pressure envelope for cuts in soft to medium clay Figure 10.7 Peck’s (1969) apparent-pressure envelope for cuts in stiff clay When using the pressure envelopes just described, keep the following points in mind: 1. 2. 3. 4. 10.3 They apply to excavations having depths greater than about 6 m. They are based on the assumption that the water table is below the bottom of the cut. Sand is assumed to be drained with zero pore water pressure. Clay is assumed to be undrained and pore water pressure is not considered. Pressure Envelope for Cuts in Layered Soil Sometimes, layers of both sand and clay are encountered when a braced cut is being constructed. In this case, Peck (1943) proposed that an equivalent value of cohesion (f 5 0) should be determined according to the formula (see Figure 10.8a). cav 5 1 3gsKsH 2s tan fsr 1 (H 2 Hs )nrqu4 2H (10.4) where H 5 total height of the cut gs 5 unit weight of sand Hs 5 height of the sand layer Ks 5 a lateral earth pressure coefficient for the sand layer (<1) fsr 5 effective angle of friction of sand qu 5 unconfined compression strength of clay nr 5 a coefficient of progressive failure (ranging from 0.5 to 1.0; average value 0.75) 10.4 Design of Various Components of a Braced Cut Sand s Hs H H1 Clay 1, c1 H2 Clay 2, c2 Hn Clay n , cn 507 H Clay c Hc (a) (b) Figure 10.8 Layered soils in braced cuts The average unit weight of the layers may be expressed as ga 5 1 3g H 1 (H 2 Hs )gc4 H s s (10.5) where gc 5 saturated unit weight of clay layer. Once the average values of cohesion and unit weight are determined, the pressure envelopes in clay can be used to design the cuts. Similarly, when several clay layers are encountered in the cut (Figure 10.8b), the average undrained cohesion becomes cav 5 1 (c H 1 c2H2 1 c 1 cnHn ) H 1 1 (10.6) where c1 , c2 , c, cn 5 undrained cohesion in layers 1, 2, c, n H1 , H2 , c, Hn 5 thickness of layers 1, 2, c, n The average unit weight is now ga 5 10.4 1 (g H 1 g2H2 1 g3H3 1 c 1 gnHn ) H 1 1 (10.7) Design of Various Components of a Braced Cut Struts In construction work, struts should have a minimum vertical spacing of about 2.75 m or more. Struts are horizontal columns subject to bending. The load-carrying capacity of columns depends on their slenderness ratio, which can be reduced by providing 508 Chapter 10: Braced Cuts vertical and horizontal supports at intermediate points. For wide cuts, splicing the struts may be necessary. For braced cuts in clayey soils, the depth of the first strut below the ground surface should be less than the depth of tensile crack, zc . From Eq. (7.8), sar 5 gzKa 2 2cr"Ka where Ka 5 coefficient of Rankine active pressure. For determining the depth of tensile crack, sar 5 0 5 gzcKa 2 2cr"Ka or zc 5 2cr "Kag With f 5 0, Ka 5 tan2 (45 2 f>2) 5 1, so zc 5 2c g A simplified conservative procedure may be used to determine the strut loads. Although this procedure will vary, depending on the engineers involved in the project, the following is a step-by-step outline of the general methodology (see Figure 10.9): Step 1. Step 2. Step 3. Draw the pressure envelope for the braced cut. (See Figures 10.5, 10.6, and 10.7.) Also, show the proposed strut levels. Figure 10.9a shows a pressure envelope for a sandy soil; however, it could also be for a clay. The strut levels are marked A, B, C, and D. The sheet piles (or soldier beams) are assumed to be hinged at the strut levels, except for the top and bottom ones. In Figure 10.9a, the hinges are at the level of struts B and C. (Many designers also assume the sheet piles or soldier beams to be hinged at all strut levels except for the top.) Determine the reactions for the two simple cantilever beams (top and bottom) and all the simple beams between. In Figure 10.9b, these reactions are A, B1 , B2 , C1 , C2 , and D. The strut loads in the figure may be calculated via the formulas PA 5 (A) (s) PB 5 (B1 1 B2 ) (s) PC 5 (C1 1 C2 ) (s) and PD 5 (D) (s) (10.8) 10.4 Design of Various Components of a Braced Cut 509 d1 A d2 B Hinges a d3 C Simple cantilever d1 d4 Section A D a d2 d5 B1 Simple beam B2 a d3 s C1 Simple cantilever C2 d4 Plan a D d5 (a) (b) Figure 10.9 Determination of strut loads: (a) section and plan of the cut; (b) method for determining strut loads where PA , PB , PC , PD 5 loads to be taken by the individual struts at levels A, B, C, and D, respectively A, B1 , B2 , C1 , C2 , D 5 reactions calculated in Step 2 (note the unit: force>unit length of the braced cut) s 5 horizontal spacing of the struts (see plan in Figure 10.9a) Step 4. Knowing the strut loads at each level and the intermediate bracing conditions allows selection of the proper sections from the steel construction manual. 510 Chapter 10: Braced Cuts Sheet Piles The following steps are involved in designing the sheet piles: Step 1. Step 2. Step 3. For each of the sections shown in Figure 10.9b, determine the maximum bending moment. Determine the maximum value of the maximum bending moments (Mmax ) obtained in Step 1. Note that the unit of this moment will be, for example, kN-m>m length of the wall. Obtain the required section modulus of the sheet piles, namely, S5 Step 4. Mmax sall (10.9) where sall 5 allowable flexural stress of the sheet pile material. Choose a sheet pile having a section modulus greater than or equal to the required section modulus from a table such as Table 9.1. Wales Wales may be treated as continuous horizontal members if they are spliced properly. Conservatively, they may also be treated as though they are pinned at the struts. For the section shown in Figure 10.9a, the maximum moments for the wales (assuming that they are pinned at the struts) are, At level A, Mmax 5 (A) (s2 ) 8 At level B, Mmax 5 (B1 1 B2 )s2 8 At level C, Mmax 5 (C1 1 C2 )s2 8 and At level D, Mmax 5 (D) (s2 ) 8 where A, B1 , B2 , C1 , C2 , and D are the reactions under the struts per unit length of the wall (see Step 2 of strut design). Now determine the section modulus of the wales: S5 Mmax sall The wales are sometimes fastened to the sheet piles at points that satisfy the lateral support requirements. 10.4 Design of Various Components of a Braced Cut 511 Example 10.1 The cross section of a long braced cut is shown in Figure 10.10a. a. b. c. d. Draw the earth-pressure envelope. Determine the strut loads at levels A, B, and C. Determine the section modulus of the sheet pile section required. Determine a design section modulus for the wales at level B. (Note: The struts are placed at 3 m, center to center, in the plan.) Use sall 5 170 3 103 kN>m2 Solution Part a We are given that g 5 18 kN>m2, c 5 35 kN>m2, and H 5 7 m. So, (18) (7) gH 5 5 3.6 , 4 c 35 Thus, the pressure envelope will be like the one in Figure 10.7. The envelope is plotted in Figure 10.10a with maximum pressure intensity, sa , equal to 0.3gH 5 0.3(18) (7) 5 37.8 kN , m2. Part b To calculate the strut loads, examine Figure 10.10b. Taking the moment about B1 , we have S MB1 5 0, and 1.75 1.75 1 ≤ 2 (1.75) (37.8) ¢ ≤ 50 A(2.5) 2 ¢ ≤ (37.8) (1.75) ¢ 1.75 1 2 3 2 or A 5 54.02 kN>m Also, S vertical forces 5 0. Thus, 1 2 (1.75) (37.8) 1 (37.8) (1.75) 5 A 1 B1 or 33.08 1 66.15 2 A 5 B1 So, B1 5 45.2 kN>m Due to symmetry, B2 5 45.2 kN>m and C 5 54.02 kN>m 512 Chapter 10: Braced Cuts 6m 1m A 1.75 m Sheet pile 2.5 m B 3.5 m 37.8 kN/m2 Clay 18 kN/m3 c 35 kN/m2 0 2.5 m C 1.75 m 1m (a) Cross section 1.75 m 1.75 m 1.75 m 37.8 kN/m2 37.8 kN/m2 1.75 m 21.6 1m 2.5 m 2.5 m A B1 1m C B2 (b) Determination of reaction 43.23 kN 43.23 kN x 1.196 m B1 A B2 E F 10.8 kN C 10.8 kN 45.2 kN 45.2 kN (c) Shear force diagram Figure 10.10 Analysis of a braced cut 10.4 Design of Various Components of a Braced Cut 513 Hence, the strut loads at the levels indicated by the subscripts are PA 5 54.02 3 horizontal spacing, s 5 54.02 3 3 5 162.06 kN PB 5 (B1 1 B2 )3 5 (45.2 1 45.2)3 5 271.2 kN and PC 5 54.02 3 3 5 162.06 kN Part c At the left side of Figure 10.10b, for the maximum moment, the shear force should be zero. The nature of the variation of the shear force is shown in Figure 10.10c. The location of point E can be given as x5 reaction at B1 45.2 5 5 1.196 m 37.8 37.8 Also, 37.8 1 1 3 1≤ ¢ ≤ Magnitude of moment at A 5 (1) ¢ 2 1.75 3 5 3.6 kN-m>meter of wall and Magnitude of moment at E 5 (45.2 3 1.196) 2 (37.8 3 1.196) ¢ 1.196 ≤ 2 5 54.06 2 27.03 5 27.03 kN-m>meter of wall Because the loading on the left and right sections of Figure 10.10b are the same, the magnitudes of the moments at F and C (see Figure 10.10c) will be the same as those at E and A, respectively. Hence, the maximum moment is 27.03 kN-m>meter of wall. The section modulus of the sheet piles is thus S5 Mmax 27.03 kN-m 5 5 15.9 3 1025m3 , m of the wall sall 170 3 103 kN>m2 Part d The reaction at level B has been calculated in part b. Hence, Mmax 5 (B1 1 B2 )s2 (45.2 1 45.2)32 5 5 101.7 kN-m 8 8 and Section modulus, S 5 101.7 101.7 5 sall (170 3 1000) 5 0.598 3 1023 m3 ■ 514 Chapter 10: Braced Cuts Example 10.2 Refer to the braced cut shown in Figure 10.11, for which ␥ ⫽ 17 kN/m3, ⬘ ⫽ 35°, and cr ⫽ 0. The struts are located 4 m on center in the plan. Draw the earth-pressure envelope and determine the strut loads at levels A, B, and C. Solution For this case, the earth-pressure envelope shown in Figure 10.5 is applicable. Hence, Ka 5 tan2 a45 2 fr 35 b 5 tan2 a45 2 b 5 0.271 2 2 From Equation (10.1) sa 5 0.65 gHKa 5 (0.65) (17) (9) (0.271) 5 26.95 kN>m2 Figure 10.12a shows the pressure envelope. Refer to Figure 10.12b and calculate B1: a MB1 5 0 5 (26.95) (5) a b 2 A5 5 112.29 kN>m 3 B1 5 (26.95) (5) 2 112.29 5 22.46 kN>m Now, refer to Figure 10.12c and calculate B2: a MB2 5 0 5m 2m A 3m B Sand c0 3m C 1m Figure 10.11 10.5 Case Studies of Braced Cuts 515 2m A 3m B a 0.65HKa 26.95 kN/m2 3m C 1m (a) 26.95 kN/m2 2m 26.95 kN/m2 3m 3m B1 A (b) B2 1m C (c) Figure 10.12 Load diagrams 4 (26.95) (4) a b 2 C5 5 71.87 kN>m 3 B2 5 (26.95) (4) 2 71.87 5 35.93 kN>m The strut loads are At A, (112.29)(spacing) ⫽ (112.29)(4) ⫽ 449.16 kN At B, (B1 ⫹ B2)(spacing) ⫽ (22.46 ⫹ 35.93)(4) ⫽ 233.56 kN At C, (71.87)(spacing) ⫽ (71.87)(4) ⫽ 287.48 kN 10.5 ■ Case Studies of Braced Cuts The procedure for determining strut loads and the design of sheet piles and wales presented in the preceding sections appears to be fairly straightforward. It is, however, only possible if a proper pressure envelope is chosen for the design, which is difficult. This section describes some case studies of braced cuts and highlights the difficulties and degree of judgment needed for successful completion of various projects. Subway Extension of the Massachusetts Bay Transportation Authority (MBTA) Lambe (1970) provided data on the performance of three excavations for the subway extension of the MBTA in Boston (test sections A, B, and D), all of which were well instrumented. Figure 10.13 gives the details of test section B, where the cut was 17.68 m, including subsoil conditions. The subsoil consisted of gravel, sand, silt, and clay (fill) to a depth of about 7.93 m, followed by a light gray, slightly organic silt to a depth of 14.02 m. A layer of coarse sand and gravel with some clay was present from 14.02 m to 16.46 m below the ground surface. Rock was encountered below 16.46 m. The horizontal spacing of the struts was 3.66 m center-to-center. 516 Chapter 10: Braced Cuts 11.28 m Strut S1 Fill 7.93 m S2 17.68 m S3 Silt 6.09 m S4 S5 2.44 m Till 1.2 m Rock Rock Figure 10.13 Schematic diagram of test section B for subway extension, MTBA Because the apparent pressure envelopes available (Section 10.2) are for sand and clay only, questions may arise about how to treat the fill, silt, and till. Figure 10.14 shows the apparent pressure envelopes proposed by Peck (1969), considering the soil as sand and also as clay, to overcome that problem. For the average soil parameters of the profile, the following values of sa were used to develop the pressure envelopes shown in Figure 10.14. 17.68 m a = 53.52 kN/m2 (a) Assuming sand a = 146.23 kN/m2 (b) Assuming clay Figure 10.14 Pressure envelopes (a) assuming sand; (b) assuming clay 10.5 Case Studies of Braced Cuts 517 Sand sa 5 0.65gHKa (10.10) For g 5 17.92 kN>m3, H 5 17.68 m, and Ka 5 0.26, sa 5 (0.65) (17.92) (17.68) (0.26) 5 53.52 kN>m2 Clay sa 5 gH c1 2 a 4c bd gH (10.11) For c 5 42.65 kN>m2, sa 5 (17.92) (17.68) c1 2 (4) (42.65) d 5 146.23 kN>m2 (17.92) (17.68) Table 10.1 shows the variations of the strut load, based on the assumed pressure envelopes shown in Figure 10.14. Also shown in Table 10.1 are the measured strut loads in the field and the design strut loads. This comparison indicates that 1. In most cases the measured strut loads differed widely from those predicted. This result is due primarily to the uncertainties involved in the assumption of the soil parameters. 2. The actual design strut loads were substantially higher than those measured. B. Construction of National Plaza (South Half) in Chicago The construction of the south half of the National Plaza in Chicago required a braced cut 21.43 m deep. Swatek et al. (1972) reported the case history for this construction. Figure 10.15 shows a schematic diagram for the braced cut and the subsoil profile. There were six levels of struts. Table 10.2 gives the actual maximum wale and strut loads. Table 10.1 Computed and Measured Strut Loads at Test Section B Computed load (kip) Strut number Envelope based on sand S-1 S-2 S-3 S-4 S-5 810 956 685 480 334 Envelope based on clay 1023 2580 1868 1299 974 Measured strut load (kip) 313 956 1352 1023 1219 518 Chapter 10: Braced Cuts 4.36 m Existing Curb wall Sand fill 30° 17.29 kN/m3 0.305 m Stiff clay 0.915 m A B Soft silty clay 0 c 19.17 kN/m2 19.97 kN/m3 C Subway D 9.76 m Medium silty clay 0 c 33.54 kN/m2, 20.44 kN/m3 13.11 m Tough silty clay 0, c 95.83 kN/m2, 21.22 kN/m3 14.94 m E F 17.07 m MZ 38 Sheet piling –18.90 m 19.51 m Very tough silty clay 0 c 191.67 kN/m2 21.22 kN/m3 Hardpan Figure 10.15 Schematic diagram of braced cut—National Plaza of Chicago Table 10.2 National Plaza Wale and Strut Loads Strut level Elevation (m) Load measured (kN/m) A B C D E F 10.915 21.83 24.57 27.47 210.37 213.57 233.49 386.71 423.20 423.20 423.20 448.0 S2337.8 Figure 10.16 presents a lateral earth-pressure envelope based on the maximum wale loads measured. To compare the theoretical prediction to the actual observation requires making an approximate calculation. To do so, we convert the clayey soil layers from Elevation 10.305 m ft to 217.07 m to a single equivalent layer in Table 10.3 by using Eq. (10.6). Now, using Eq. (10.4), we can convert the sand layer located between elevations 14.36 m and 10.305 m and the equivalent clay layer of 17.375 m to one equivalent clay layer with a thickness of 21.43 m: 1 3gsKsH 2s tan fsr 1 (H 2 Hs )nrqu4 2H 1 d 3 (17.29) (1) (4.055) 2 tan 30 1 (17.375) (0.75) (2 3 51.24) 4 5 c (2) (21.43) < 34.99 kN>m2 cav 5 10.5 Case Studies of Braced Cuts +4.36 m 5.36 m +0.915 m A –1.83 m B –4.57 m C 283.7 kN/m2 –7.47 m D Peck's pressure envelope –10.37 m E 16.07 m Actual pressure envelope –13.57 m F Bottom of cut –17.07 m Figure 10.16 Comparison of actual and Peck’s pressure envelopes Table 10.3 Conversion of Soil Layers using Eq. (10.6) Elevation (m) Thickness, H (m) c (kN/m2) 10.305 to 29.67 9.975 19.17 29.67 to 213.11 3.44 33.54 213.11 to 214.94 214.94 to 217.07 1.83 2.13 S17.375 Equivalent c (kN/m2) 1 3 (9.975) (19.17) 1 (3.44) (33.54) 17.375 1 (1.83) (95.83) (2.13) (191.67) 5 51.24 kN>m2 cav 5 95.83 191.67 Equation (10.7) gives 1 gav 5 3g H 1 g2H2 1 c 1 gnHn ) H 1 1 519 520 Chapter 10: Braced Cuts 1 3(17.29) (4.055) 1 (19.97) (10.065) 1 (20.44) (3.35) 21.43 1 (21.22) (1.83) 1 (21.22) (2.13)4 5 19.77 kN>m3 5 For the equivalent clay layer of 21.43 m, gavH (19.77) (21.43) 5 5 12.1 . 4 cav 34.99 Hence the apparent pressure envelope will be of the type shown in Figure 10.6 From Eq. (10.2) 4cav (4) (34.99) sa 5 gH c1 2 a bd 5 (19.77) (21.43) c1 2 d 5 283.7 kN>m2 gavH (19.77) (12.43) The pressure envelope is shown in Figure 10.16. The area of this pressure diagram is 2933 kN/m. Thus Peck’s pressure envelope gives a lateral earth pressure of about 1.8 times that actually observed. This result is not surprising because the pressure envelope provided by Figure 10.6 is an envelope developed considering several cuts made at different locations. Under actual field conditions, past experience with the behavior of similar soils can help reduce overdesigning substantially. 10.6 Bottom Heave of a Cut in Clay Braced cuts in clay may become unstable as a result of heaving of the bottom of the excavation. Terzaghi (1943) analyzed the factor of safety of long braced excavations against bottom heave. The failure surface for such a case in a homogeneous soil is shown in Figure 10.17. In the figure, the following notations are used: B 5 width of the cut, H 5 depth of the cut, T 5 thickness of the clay below the base of excavation, and q 5 uniform surcharge adjacent to the excavation. q e j B c B 0 c H B g f 45° i 45° T h Arc of a circle Figure 10.17 Heaving in braced cuts in clay 10.6 Bottom Heave of a Cut in Clay 521 The ultimate bearing capacity at the base of a soil column with a width of Br can be given as qult 5 cNc (10.12) where Nc 5 5.7 (for a perfectly rough foundation). The vertical load per unit area along fi is q 5 gH 1 q 2 cH Br (10.13) Hence, the factor of safety against bottom heave is FS 5 qult 5 q cNc cH gH 1 q 2 Br 5 cNc q c ¢g 1 2 ≤H H Br (10.14) For excavations of limited length L, the factor of safety can be modified to cNc ¢1 1 0.2 FS 5 Br ≤ L q c ¢g 1 2 ≤H H Br (10.15) where Br 5 T or B>"2 (whichever is smaller). In 2000, Chang suggested a revision of Eq. (10.15) with the following changes: 1. The shearing resistance along ij may be considered as an increase in resistance rather than a reduction in loading. 2. In Figure 10.17, fg with a width of Bs at the base of the excavation may be treated as a negatively loaded footing. 3. The value of the bearing capacity factor Nc should be 5.14 (not 5.7) for a perfectly smooth footing, because of the restraint-free surface at the base of the excavation. With the foregoing modifications, Eq. (10.15) takes the form 5.14c ¢1 1 FS 5 0.2Bs cH ≤ 1 L Br gH 1 q (10.16) where Br 5 T if T < B>"2 Br 5 B>"2 if T . B>"2 Bs 5 "2Br Bjerrum and Eide (1956) compiled a number of case records for the bottom heave of cuts in clay. Chang (2000) used those records to calculate FS by means of Eq. (10.16); his findings are summarized in Table 10.4. It can be seen from this table that the actual field observations agree well with the calculated factors of safety. 522 Chapter 10: Braced Cuts Table 10.4 Calculated Factors of Safety for Selected Case Records Compiled by Bjerrum and Eide (1956) and Calculated by Chang (2000) Site Pumping station, Fornebu, Oslo Storehouse, Drammen Sewerage tank, Drammen Excavation, Grey Wedels Plass, Oslo Pumping station, Jernbanetorget, Oslo Storehouse, Freia, Oslo Subway, Chicago B (m) B,L H (m) H,B g (kN , m3) 5.0 1.0 3.0 0.6 17.5 4.8 0 2.4 0.5 19.0 5.5 0.69 3.5 0.64 5.8 0.72 4.5 8.5 0.70 5.0 16 0 0 c (kN , m2) q (kN , m2) FS [Eq. (10.16)] 0 1.05 Total failure 12 15 1.05 Total failure 18.0 10 10 0.92 Total failure 0.78 18.0 14 10 1.07 Total failure 6.3 0.74 19.0 22 0 1.26 Partial failure 5.0 11.3 1.00 0.70 19.0 19.0 16 35 0 0 1.10 1.00 Partial failure Near failure 7.5 Type of failure Equation (10.16) is recommended for use in this test. In most cases, a factor of safety of about 1.5 is recommended. In homogeneous clay, if FS becomes less than 1.5, the sheet pile is driven deeper. (See Figure 10.18.) Usually, the depth d is kept less than or equal to B>2, in which case the force P per unit length of the buried sheet pile (aar and bbr) may be expressed as (U.S. Department of the Navy, 1971) P 5 0.7(gHB 2 1.4cH 2 pcB) for d . 0.47B (10.17) B c 0 H d a b a b P P Figure 10.18 Force on the buried length of sheet pile 10.6 Bottom Heave of a Cut in Clay 523 and P 5 1.5d ¢gH 2 1.4cH 2 pc ≤ B for d , 0.47B (10.18) Example 10.3 In Figure 10.19. for a braced cut in clay, B 5 3 m, L 5 20 m, H 5 5.5 m, T 5 1.5 m, g 5 17 kN>m3, c 5 30 kN>m2, and q 5 0. Calculate the factor of safety against heave. Use Eq. (10.16). Solution From Eq. (10.16), 0.2Brr cH b1 L Br gH 1 q 5.14ca1 1 FS 5 with T 5 1.5 m, B !2 So 5 3 !2 T# Hence, Br 5 T 5 1.5 m, and it follows that 5 2.12 m B !2 B rr 5 !2Br 5 ( !2) (1.5) 5 2.12 m 3m Clay γ = 17 kN/m3 5.5 m c = 30 kN/m3 φ=0 1.5 m Hard stratum Figure 10.19 Factor of safety against heaving for a braced cut and FS 5 (5.14) (30) c1 1 (0.2) (2.12) (30) (5.5) d 1 20 1.5 5 2.86 (17) (5.5) ■ 524 Chapter 10: Braced Cuts 10.7 Stability of the Bottom of a Cut in Sand The bottom of a cut in sand is generally stable. When the water table is encountered, the bottom of the cut is stable as long as the water level inside the excavation is higher than the groundwater level. In case dewatering is needed (see Figure 10.20), the factor of safety against piping should be checked. [Piping is another term for failure by heave, as defined in Section 1.12; see Eq. (1.45).] Piping may occur when a high hydraulic gradient is created by water flowing into the excavation. To check the factor of safety, draw flow nets and determine the maximum exit gradient 3imax(exit) 4 that will occur at points A and B. Figure 10.21 shows such a flow net, for which the maximum exit gradient is imax(exit) h Nd h 5 5 a Nda (10.19) where a 5 length of the flow element at A (or B) Nd 5 number of drops (Note: in Figure 10.21, Nd 5 8; see also Section 1.11) The factor of safety against piping may be expressed as FS 5 icr imax(exit) (10.20) where icr 5 critical hydraulic gradient. Water level Water level B h L1 A B L2 Flow of water Impervious layer L3 Figure 10.20 Stability of the bottom of a cut in sand 10.7 Stability of the Bottom of a Cut in Sand Water table 525 Water table h A Water level B a 1 8 7 2 6 5 3 4 Impervious layer Figure 10.21 Determining the factor of safety against piping by drawing a flow net The relationship for icr was given in Chapter 1 as icr 5 Gs 2 1 e11 The magnitude of icr varies between 0.9 and 1.1 in most soils, with an average of about 1. A factor of safety of about 1.5 is desirable. The maximum exit gradient for sheeted excavations in sands with L3 5 ` can also be evaluated theoretically (Harr, 1962). (Only the results of these mathematical derivations will be presented here. For further details, see the original work.) To calculate the maximum exit gradient, examine Figures 10.22 and 10.23 and perform the following steps: 1. Determine the modulus, m, from Figure 10.22 by obtaining 2L2>B (or B>2L2) and 2L1>B. 2. With the known modulus and 2L1>B, examine Figure 10.23 and determine L2iexit(max)>h. Because L2 and h will be known, iexit(max) can be calculated. 3. The factor of safety against piping can be evaluated by using Eq. (10.20). Marsland (1958) presented the results of model tests conducted to study the influence of seepage on the stability of sheeted excavations in sand. The results were summarized by the U.S. Department of the Navy (1971) in NAVFAC DM-7 and are given in Figure 10.24a, b, and c. Note that Figure 10.24b is for the case of determining the sheet pile penetration L2 needed for the required factor of safety against piping when the sand layer extends to a great depth below the excavation. By contrast, Figure 10.24c represents the case in which an impervious layer lies at a limited depth below the bottom of the excavation. 526 Chapter 10: Braced Cuts Text not available due to copyright restrictions 10.7 Stability of the Bottom of a Cut in Sand 0.70 0.65 L2i exit(max) h 0.60 2L1 B 0.55 = 0 0.5 0.50 1 2 4 8 12 16 20 0.45 0.40 0 0.02 0.04 0.06 0.08 Modulus, m (a) 0.10 0.12 1.0 1.2 0.6 0.5 L2i exit(max) h 0.4 2L1 =0 B 0.3 0.5 0.2 1 0.1 0 2 20 12 8 4 16 0.2 0.4 0.6 0.8 Modulus, m (b) Figure 10.23 Variation of maximum exit gradient with modulus (From Groundwater and Seepage, by M. E. Harr. Copyright 1962 by McGraw-Hill. Used with permission.) 527 528 Chapter 10: Braced Cuts B Water table h Sand L2 L3 Impervious layer (a) 2.0 Loose sand Dense sand Factor of safety aganist heave in loose sand or piping in dense sand 2.0 L3 = ∞ 1.5 L2 1.0 h 1.5 2.0 1.5 1.0 1.0 0.5 0 0 0.5 1.0 B/2h (b) 2.0 2.0 1.5 Dense sand of limited depth: L3 1.5 L2 1.0 h L3 =2 h Factors of safety agnist piping 2.0 0.5 2.0 1.5 1.5 1.0 1.0 L3 =1 h 0 0 0.5 1.0 1.5 2.0 B/2h (c) Figure 10.24 Influence of seepage on the stability of sheeted excavation (US Department of Navy, 1971.) 10.8 Lateral Yielding of Sheet Piles and Ground Settlement 529 Example 10.4 In Figure 10.20, let h 5 4.5 m, L1 5 5 m, L2 5 4 m, B 5 5 m, and L3 5 ` . Determine the factor of safety against piping. Use Figures 10.22 and 10.23. Solution We have 2L1 2(5) 5 52 B 5 and B 5 5 5 0.625 2L2 2(4) According to Figure 10.22b, for 2L1>B 5 2 and B>2L2 5 0.625,m < 0.033. From Figure 10.23a, for m 5 0.033 and 2L1>B 5 2, L2iexit(max)>h 5 0.54. Hence, iexit(max) 5 0.54(h) 5 0.54(4.5)>4 5 0.608 L2 and FS 5 10.8 icr 1 5 5 1.645 i max (exit) 0.608 ■ Lateral Yielding of Sheet Piles and Ground Settlement In braced cuts, some lateral movement of sheet pile walls may be expected. (See Figure 10.25.) The amount of lateral yield (dH ) depends on several factors, the most important of which is the elapsed time between excavation and the placement of wales and struts. As discussed before, in several instances the sheet piles (or the soldier piles, as the case may be) are driven to a certain depth below the bottom of the excavation. The reason is to reduce the lateral yielding of the walls during the last stages of excavation. Lateral yielding of the walls will cause the ground surface surrounding the cut to settle. The degree of lateral yielding, however, depends mostly on the type of soil below the bottom of the cut. If clay below the cut extends to a great depth and gH>c is less than about 6, extension of the sheet piles or soldier piles below the bottom of the cut will help considerably in reducing the lateral yield of the walls. 530 Chapter 10: Braced Cuts x Original ground surface V (max) Deflected shape of sheet pile z z H H H (max) Figure 10.25 Lateral yielding of sheet pile and ground settlement However, under similar circumstances, if gH>c is about 8, the extension of sheet piles into the clay below the cut does not help greatly. In such circumstances, we may expect a great degree of wall yielding that could result in the total collapse of the bracing systems. If a hard layer of soil lies below a clay layer at the bottom of the cut, the piles should be embedded in the stiffer layer. This action will greatly reduce lateral yield. The lateral yielding of walls will generally induce ground settlement, dV , around a braced cut. Such settlement is generally referred to as ground loss. On the basis of several field observations, Peck (1969) provided curves for predicting ground settlement in various types of soil. (See Figure 10.26.) The magnitude of ground loss varies extensively; however, the figure may be used as a general guide. Moormann (2004) analyzed about 153 case histories dealing mainly with the excavation in soft clay (that is, undrained shear strength, c < 75 kN>m2). Following is a summary of his analysis relating to dV(max), x r, dH(max), and z r (see Figure 10.25). • Maximum Vertical Movement [dV(max)] dV(max)>H < 0.1 to 10.1% with an average of 1.07% (soft clay) dV(max)>H < 0 to 0.9% with an average of 0.18% (stiff clay) dV(max)>H < 0 to 2.43% with an average of 0.33% (non-cohesive soils) • Location of dV(max), that is x r (Figure 10.25) For 70% of all case histories considered, x r < 0.5H. However, in soft clays, x r may be as much as 2H. Problems 531 3 A — Sand and soft clay and average workmanship B — Very soft to soft clay. Limited in depth below base of excavation 2 C — Very soft to soft clay. Great depth below excavation V (%) H 1 C B A 0 1 2 3 Distance from the braced wall H 4 Figure 10.26 Variation of ground settlement with distance (From Peck, R. B. (1969). “Deep Excavation and Tunneling in Soft Ground,” Proceedings Seventh International Conference on Soil Mechanics and Foundation Engineering, Mexico City, State-of-the-Art Volume, pp. 225–290. With permission from ASCE.) • Maximum Horizontal Deflection of Sheet Piles, dH(max) For 40% of excavation in soft clay, 0.5% < dH(max)>H < 1%. The average value of dH(max)>H is about 0.87%. In stiff clays, the average value of dH(max)>H is about 0.25%. In non-cohesive soils, dH(max)>H is about 0.27% of the average. • Location of dH(max), that is z r (Figure 10.25) For deep excavation of soft and stiff cohesive soils, zr>H is about 0.5 to 1.0. Problems 10.1 Refer to the braced cut shown in Figure P10.1. Given: g 5 16 kN>m3, fr 5 38°, and c r 5 0. The struts are located at 3.5 m center-to-center in the plan. Draw the earth-pressure envelope and determine the strut loads at levels A, B, and C. 532 Chapter 10: Braced Cuts 4.5 m 1m A 2.5 m B Sand 16 kN/m3 38° c 0 3m C 1.5 m Figure P10.1 10.2 For the braced cut described in Problem 10.1, determine the following: a. The sheet-pile section modulus b. The section modulus of the wales at level B Assume that sall 5 170 MN>m2. 10.3 Refer to Fig. P.10.3. Redo Problem 10.1 with g 5 18 kN>m3, fr 5 40°, c r 5 0, and the center-to-center strut spacing in the plan 5 4 m. 10.4 Determine the sheet-pile section modulus for the braced cut described in Problem 10.3. Given: sall 5 170 MN>m2. 10.5 Refer to Figure 10.8a. For the braced cut, given H 5 6 m; Hs 5 2.5 m; gs 5 16.5 kN>m3; angle of friction of sand, fsr 5 35°; Hc 5 3.5 m; gc 5 17.5 kN>m3; and unconfined compression strength of clay layer, qu 5 62 kN>m2. a. Estimate the average cohesion (cav ) and average unit weight (gav ) for the construction of the earth-pressure envelope. b. Plot the earth-pressure envelope. 3.5 m 1m A 2m B 2m C 1.5 m Figure P10.3 Sand 18 kN/m3 38° c 0 References 533 6m 1m A 3m B c 30 kN/m2 0 17.5 kN/m3 2m C 1m Figure P10.7 10.6 Refer to Figure 10.8b, which shows a braced cut in clay. Given: H 5 7.6 m, H1 5 1.52 m, c1 5 101.8 kN>m2, g1 5 17.45 kN>m3, H2 5 3.04 m, c2 5 74.56 kN>m2, g2 5 16.83 kN>m3, H3 5 3.04 m, c3 5 80.02 kN>m2, and g3 5 17.14 kN>m3. a. Determine the average cohesion (cav ) and average unit weight (gav ) for the construction of the earth-pressure envelope. b. Plot the earth-pressure envelope. 10.7 Refer to Figure P10.7. Given: g 5 17.5 kN>m3, c 5 30 kN>m2, and center-tocenter spacing of struts in the plan 5 5 m. Draw the earth-pressure envelope and determine the strut loads at levels A, B, and C. 10.8 Determine the sheet-pile section modulus for the braced cut described in Problem 10.7. Use sall 5 170 MN>m.2. 10.9 Redo Problem 10.7 assuming that c 5 60 kN>m2. 10.10 Determine the factor of safety against bottom heave for the braced cut described in Problem 10.7. Use Eq. (10.16) and assume the length of the cut, L 5 18 m. 10.11 Determine the factor of safety against bottom heave for the braced cut described in Problem 10.9. Use Eq. (10.15). The length of the cut is 12.5. References BJERRUM, L, and EIDE, O. (1956). “Stability of Strutted Excavation in Clay,” Geotechnique, Vol. 6, No. 1, pp. 32–47. CHANG, M. F. (2000). “Basal Stability Analysis of Braced Cuts in Clay,” Journal of Geotechnical and Geoenvironmental Engineering, ASCE, Vol. 126, No. 3, pp. 276–279. HARR, M. E. (1962). Groundwater and Seepage, McGraw-Hill, New York. LAMBE, T. W. (1970). “Braced Excavations.” Proceedings of the Specialty Conference on Lateral Stresses in the Ground and Design of Earth-Retaining Structures, American Society of Civil Engineers, pp. 149–218. 534 Chapter 10: Braced Cuts MOORMANN, C. (2004). “Analysis of Wall and Ground Movements Due to Deep Excavations in Soft Soil Based on New Worldwide Data Base,” Soils and Foundations, Vol. 44, No. 1, pp. 87–98. PECK, R. B. (1943). “Earth Pressure Measurements in Open Cuts, Chicago (ILL.) Subway,” Transactions, American Society of Civil Engineers, Vol. 108, pp. 1008–1058. PECK, R. B. (1969). “Deep Excavation and Tunneling in Soft Ground,” Proceedings Seventh International Conference on Soil Mechanics and Foundation Engineering, Mexico City, State-of-theArt Volume, pp. 225–290. SWATEK, E. P., JR., ASROW, S. P., and SEITZ, A. (1972). “Performance of Bracing for Deep Chicago Excavation,” Proceeding of the Specialty Conference on Performance of Earth and Earth Supported Structures, American Society of Civil Engineers, Vol. 1, Part 2, pp. 1303–1322. TERZAGHI, K. (1943). Theoretical Soil Mechanics, Wiley, New York. U.S. DEPARTMENT OF THE NAVY (1971). “Design Manual—Soil Mechanics. Foundations, and Earth Structures.” NAVFAC DM-7, Washington, D.C. 11 11.1 Pile Foundations Introduction Piles are structural members that are made of steel, concrete, or timber. They are used to build pile foundations, which are deep and which cost more than shallow foundations. (See Chapters 3, 4, and 5.) Despite the cost, the use of piles often is necessary to ensure structural safety. The following list identifies some of the conditions that require pile foundations (Vesic, 1977): 1. When one or more upper soil layers are highly compressible and too weak to support the load transmitted by the superstructure, piles are used to transmit the load to underlying bedrock or a stronger soil layer, as shown in Figure 11.1a. When bedrock is not encountered at a reasonable depth below the ground surface, piles are used to transmit the structural load to the soil gradually. The resistance to the applied structural load is derived mainly from the frictional resistance developed at the soil–pile interface. (See Figure 11.1b.) 2. When subjected to horizontal forces (see Figure 11.1c), pile foundations resist by bending, while still supporting the vertical load transmitted by the superstructure. This type of situation is generally encountered in the design and construction of earth-retaining structures and foundations of tall structures that are subjected to high wind or to earthquake forces. 3. In many cases, expansive and collapsible soils may be present at the site of a proposed structure. These soils may extend to a great depth below the ground surface. Expansive soils swell and shrink as their moisture content increases and decreases, and the pressure of the swelling can be considerable. If shallow foundations are used in such circumstances, the structure may suffer considerable damage. However, pile foundations may be considered as an alternative when piles are extended beyond the active zone, which is where swelling and shrinking occur. (See Figure 11.1d) Soils such as loess are collapsible in nature. When the moisture content of these soils increases, their structures may break down. A sudden decrease in the void ratio of soil induces large settlements of structures supported by shallow foundations. In such 535 536 Chapter 11: Pile Foundations Rock (a) (b) (c) Zone of erosion Swelling soil Stable soil (d) (e) (f) Figure 11.1 Conditions that require the use of pile foundations cases, pile foundations may be used in which the piles are extended into stable soil layers beyond the zone where moisture will change. 4. The foundations of some structures, such as transmission towers, offshore platforms, and basement mats below the water table, are subjected to uplifting forces. Piles are sometimes used for these foundations to resist the uplifting force. (See Figure 11.1e.) 5. Bridge abutments and piers are usually constructed over pile foundations to avoid the loss of bearing capacity that a shallow foundation might suffer because of soil erosion at the ground surface. (See Figure 11.1f.) Although numerous investigations, both theoretical and experimental, have been conducted in the past to predict the behavior and the load-bearing capacity of piles in granular and cohesive soils, the mechanisms are not yet entirely understood and may never be. The design and analysis of pile foundations may thus be considered somewhat of an art as a result of the uncertainties involved in working with some subsoil conditions. This chapter discusses the present state of the art. 11.2 Types of Piles and Their Structural Characteristics 11.2 537 Types of Piles and Their Structural Characteristics Different types of piles are used in construction work, depending on the type of load to be carried, the subsoil conditions, and the location of the water table. Piles can be divided into the following categories: (a) steel piles, (b) concrete piles, (c) wooden (timber) piles, and (d) composite piles. Steel Piles Steel piles generally are either pipe piles or rolled steel H-section piles. Pipe piles can be driven into the ground with their ends open or closed. Wide-flange and I-section steel beams can also be used as piles. However, H-section piles are usually preferred because their web and flange thicknesses are equal. (In wide-flange and I-section beams, the web thicknesses are smaller than the thicknesses of the flange.) Table 11.1 gives the dimensions of some standard H-section steel piles used in the United States. Table 11.2 shows selected pipe sections frequency used for piling purposes. In many cases, the pipe piles are filled with concrete after they have been driven. The allowable structural capacity for steel piles is Qall 5 A sfs (11.1) where A s 5 cross-sectional area of the steel fs 5 allowable stress of steel (<0.33–0.5 f y) Once the design load for a pile is fixed, one should determine, on the basis of geotechnical considerations, whether Q(design) is within the allowable range as defined by Eq. 11.1. When necessary, steel piles are spliced by welding or by riveting. Figure 11.2a shows a typical splice by welding for an H-pile. A typical splice by welding for a pipe pile is shown in Figure 11.2b. Figure 11.2c is a diagram of a splice of an H-pile by rivets or bolts. When hard driving conditions are expected, such as driving through dense gravel, shale, or soft rock, steel piles can be fitted with driving points or shoes. Figures 11.2d and 11.2e are diagrams of two types of shoe used for pipe piles. Steel piles may be subject to corrosion. For example, swamps, peats, and other organic soils are corrosive. Soils that have a pH greater than 7 are not so corrosive. To offset the effect of corrosion, an additional thickness of steel (over the actual designed cross-sectional area) is generally recommended. In many circumstances factory-applied epoxy coatings on piles work satisfactorily against corrosion. These coatings are not easily damaged by pile driving. Concrete encasement of steel piles in most corrosive zones also protects against corrosion. Following are some general facts about steel piles: • • Usual length: 15 m to 60 m Usual load: 300 kN to 1200 kN 538 Chapter 11: Pile Foundations • • Advantages: a. Easy to handle with respect to cutoff and extension to the desired length b. Can stand high driving stresses c. Can penetrate hard layers such as dense gravel and soft rock d. High load-carrying capacity Disadvantages: a. Relatively costly b. High level of noise during pile driving c. Subject to corrosion d. H-piles may be damaged or deflected from the vertical during driving through hard layers or past major obstructions Table 11.1 Common H-Pile Sections used in the United States Moment of inertia (m4 3 1026) Designation, size (mm) 3 weight (kg/m) Depth d1 (mm) Section area (m2 3 1023) Flange and web thickness w (mm) Flange width d2 (mm) lxx HP 200 3 53 HP 250 3 85 3 62 HP 310 3 125 3 110 3 93 3 79 HP 330 3 149 3 129 3 109 3 89 HP 360 3 174 3 152 3 132 3 108 204 254 246 312 308 303 299 334 329 324 319 361 356 351 346 6.84 10.8 8.0 15.9 14.1 11.9 10.0 19.0 16.5 13.9 11.3 22.2 19.4 16.8 13.8 11.3 14.4 10.6 17.5 15.49 13.1 11.05 19.45 16.9 14.5 11.7 20.45 17.91 15.62 12.82 207 260 256 312 310 308 306 335 333 330 328 378 376 373 371 49.4 123 87.5 271 237 197 164 370 314 263 210 508 437 374 303 y d2 d1 x x w w y lyy 16.8 42 24 89 77.5 63.7 62.9 123 104 86 69 184 158 136 109 11.2 Types of Piles and Their Structural Characteristics Table 11.2 Selected Pipe Pile Sections Outside diameter (mm) Wall thickness (mm) 219 3.17 21.5 4.78 32.1 5.56 37.3 7.92 52.7 4.78 37.5 5.56 43.6 6.35 49.4 4.78 44.9 5.56 52.3 6.35 59.7 4.78 60.3 5.56 70.1 6.35 79.8 5.56 80 6.35 90 7.92 112 5.56 88 6.35 100 7.92 125 6.35 121 7.92 150 9.53 179 12.70 238 254 305 406 457 508 610 Area of steel (cm2) 539 540 Chapter 11: Pile Foundations Weld Weld (b) (a) (c) Weld Weld (d) (e) Figure 11.2 Steel piles: (a) splicing of H-pile by welding; (b) splicing of pipe pile by welding; (c) splicing of H-pile by rivets and bolts; (d) flat driving point of pipe pile; (e) conical driving point of pipe pile Concrete Piles Concrete piles may be divided into two basic categories: (a) precast piles and (b) cast-in-situ piles. Precast piles can be prepared by using ordinary reinforcement, and they can be square or octagonal in cross section. (See Figure 11.3.) Reinforcement is provided to enable the pile to resist the bending moment developed during pickup and transportation, the vertical load, and the bending moment caused by a lateral load. The piles are cast to desired lengths and cured before being transported to the work sites. Some general facts about concrete piles are as follows: • • • Usual length: 10 m to 15 m Usual load: 300 kN to 3000 kN Advantages: a. Can be subjected to hard driving b. Corrosion resistant c. Can be easily combined with a concrete superstructure 11.2 Types of Piles and Their Structural Characteristics 541 2D Square pile Octagonal pile D D Figure 11.3 Precast piles with ordinary reinforcement • Disadvantages: a. Difficult to achieve proper cutoff b. Difficult to transport Precast piles can also be prestressed by the use of high-strength steel pre-stressing cables. The ultimate strength of these cables is about 1800 MN>m2. During casting of the piles, the cables are pretensioned to about 900 to 1300 MN>m2, and concrete is poured around them. After curing, the cables are cut, producing a compressive force on the pile section. Table 11.3 gives additional information about prestressed concrete piles with square and octagonal cross sections. Some general facts about precast prestressed piles are as follows: • • • Usual length: 10 m to 45 m Maximum length: 60 m Maximum load: 7500 kN to 8500 kN The advantages and disadvantages are the same as those of precast piles. Cast-in-situ, or cast-in-place, piles are built by making a hole in the ground and then filling it with concrete. Various types of cast-in-place concrete piles are currently used in construction, and most of them have been patented by their manufacturers. These piles may be divided into two broad categories: (a) cased and (b) uncased. Both types may have a pedestal at the bottom. Cased piles are made by driving a steel casing into the ground with the help of a mandrel placed inside the casing. When the pile reaches the proper depth the mandrel is withdrawn and the casing is filled with concrete. Figures 11.4a, 11.4b, 11.4c, and 11.4d show some examples of cased piles without a pedestal. Figure 11.4e shows a cased pile with a pedestal. The pedestal is an expanded concrete bulb that is formed by dropping a hammer on fresh concrete. Some general facts about cased cast-in-place piles are as follows: • • • • Usual length: 5 m to 15 m Maximum length: 30 m to 40 m Usual load: 200 kN to 500 kN Approximate maximum load: 800 kN 542 Chapter 11: Pile Foundations Table 11.3 Typical Prestressed Concrete Pile in Use Design bearing capacity (kN) Pile shapea D (mm) Area of cross section (cm2) Perimeter (mm) 12.7-mm diameter 11.1-mm diameter Minimum effective prestress force (kN) S O S O S O S O S O S O S O S O 254 254 305 305 356 356 406 406 457 457 508 508 559 559 610 610 645 536 929 768 1265 1045 1652 1368 2090 1729 2581 2136 3123 2587 3658 3078 1016 838 1219 1016 1422 1168 1626 1346 1829 1524 2032 1677 2235 1854 2438 2032 4 4 5 4 6 5 8 7 10 8 12 10 15 12 18 15 4 4 6 5 8 7 11 9 13 11 16 14 20 16 23 19 312 258 449 369 610 503 796 658 1010 836 1245 1032 1508 1250 1793 1486 a Number of strands Strength of concrete (MN/m2) Section modulus (m3 3 10 23) 34.5 41.4 2.737 1.786 4.719 3.097 7.489 4.916 11.192 7.341 15.928 10.455 21.844 14.355 29.087 19.107 37.756 34.794 556 462 801 662 1091 901 1425 1180 1803 1491 2226 1842 2694 2231 3155 2655 778 555 962 795 1310 1082 1710 1416 2163 1790 2672 2239 3232 2678 3786 3186 S 5 square section; O 5 octagonal section Wire spiral Prestressed strand D D Wire spiral • • • Prestressed strand Advantages: a. Relatively cheap b. Allow for inspection before pouring concrete c. Easy to extend Disadvantages: a. Difficult to splice after concreting b. Thin casings may be damaged during driving Allowable load: Qall 5 A sfs 1 A cfc where A s 5 area of cross section of steel A c 5 area of cross section of concrete fs 5 allowable stress of steel fc 5 allowable stress of concrete (11.2) 11.2 Types of Piles and Their Structural Characteristics Monotube or Union Metal Pile Western Cased Pile Thin, fluted, tapered steel casing driven without mandrel Thin metal casing Raymond Step-Taper Pile Corrugated thin cylindrical casing Maximum usual length: 30 m (a) Maximum usual length: 30 m –40 m Maximum usual length: 40 m (b) (c) Seamless Pile or Armco Pile Franki Cased Pedestal Pile Thin metal casing Straight steel pile casing Maximum usual length: 30 m– 40 m (d) Maximum usual length: 30 m– 40 m (e) Western Uncased Pile without Pedestal Franki Uncased Pedestal Pile Maximum usual length: 30 m –40 m Maximum usual length: 15 m –20 m (f) Figure 11.4 Cast-in-place concrete piles 543 (g) 544 Chapter 11: Pile Foundations Figures 11.4f and 11.4g are two types of uncased pile, one with a pedestal and the other without. The uncased piles are made by first driving the casing to the desired depth and then filling it with fresh concrete. The casing is then gradually withdrawn. Following are some general facts about uncased cast-in-place concrete piles: • • • • • • • Usual length: 5 m to 15 m Maximum length: 30 m to 40 m Usual load: 300 kN to 500 kN Approximate maximum load: 700 kN Advantages: a. Initially economical b. Can be finished at any elevation Disadvantages: a. Voids may be created if concrete is placed rapidly b. Difficult to splice after concreting c. In soft soils, the sides of the hole may cave in, squeezing the concrete Allowable load: Qall 5 A cfc (11.3) where A c 5 area of cross section of concrete fc 5 allowable stress of concrete Timber Piles Timber piles are tree trunks that have had their branches and bark carefully trimmed off. The maximum length of most timber piles is 10 to 20 m. To qualify for use as a pile, the timber should be straight, sound, and without any defects. The American Society of Civil Engineers’ Manual of Practice, No. 17 (1959), divided timber piles into three classes: 1. Class A piles carry heavy loads. The minimum diameter of the butt should be 356 mm. 2. Class B piles are used to carry medium loads. The minimum butt diameter should be 305 to 330 mm. 3. Class C piles are used in temporary construction work. They can be used permanently for structures when the entire pile is below the water table. The minimum butt diameter should be 305 mm. In any case, a pile tip should not have a diameter less than 150 mm. Timber piles cannot withstand hard driving stress; therefore, the pile capacity is generally limited. Steel shoes may be used to avoid damage at the pile tip (bottom). The tops of timber piles may also be damaged during the driving operation. The crushing of the wooden fibers caused by the impact of the hammer is referred to as brooming. To avoid damage to the top of the pile, a metal band or a cap may be used. Splicing of timber piles should be avoided, particularly when they are expected to carry a tensile load or a lateral load. However, if splicing is necessary, it can be done by using pipe sleeves (see Figure 11.5a) or metal straps and bolts (see Figure 11.5b). The length of the sleeve should be at least five times the diameter of the pile. The butting ends should be cut square so that full contact can be maintained. The spliced portions should be carefully trimmed so that they fit tightly to the inside of the pipe sleeve. In the case of metal straps and bolts, the butting ends should also be cut square. The sides of the spliced portion should be trimmed plane for putting the straps on. 11.2 Types of Piles and Their Structural Characteristics 545 Metal strap Metal sleeve Ends cut square Ends cut square Metal strap (a) (b) Figure 11.5 Splicing of timber piles: (a) use of pipe sleeves; (b) use of metal straps and bolts Timber piles can stay undamaged indefinitely if they are surrounded by saturated soil. However, in a marine environment, timber piles are subject to attack by various organisms and can be damaged extensively in a few months. When located above the water table, the piles are subject to attack by insects. The life of the piles may be increased by treating them with preservatives such as creosote. The allowable load-carrying capacity of wooden piles is Qall 5 A pfw (11.4) where A p 5 average area of cross section of the pile fw 5 allowable stress on the timber The following allowable stresses are for pressure-treated round timber piles made from Pacific Coast Douglas fir and Southern pine used in hydraulic structures (ASCE, 1993): Pacific Coast Douglas Fir • • • • Compression parallel to grain: 6.04 MN>m2 Bending: 11.7 MN>m2 Horizontal shear: 0.66 MN>m2 Compression perpendicular to grain: 1.31 MN>m2 Southern Pine • Compression parallel to grain: 5.7 MN>m2 • Bending: 11.4 MN>m2 546 Chapter 11: Pile Foundations • Horizontal shear: 0.62 MN>m2 • Compression perpendicular to grain: 1.41 MN>m2 The usual length of wooden piles is 5 m to 15 m. The maximum length is about 30 m to 40 m (100 ft to 130 ft). The usual load carried by wooden piles is 300 kN to 500 kN. Composite Piles The upper and lower portions of composite piles are made of different materials. For example, composite piles may be made of steel and concrete or timber and concrete. Steel-and-concrete piles consist of a lower portion of steel and an upper portion of castin-place concrete. This type of pile is used when the length of the pile required for adequate bearing exceeds the capacity of simple cast-in-place concrete piles. Timber-andconcrete piles usually consist of a lower portion of timber pile below the permanent water table and an upper portion of concrete. In any case, forming proper joints between two dissimilar materials is difficult, and for that reason, composite piles are not widely used. 11.3 Estimating Pile Length Selecting the type of pile to be used and estimating its necessary length are fairly difficult tasks that require good judgment. In addition to being broken down into the classification given in Section 11.2, piles can be divided into three major categories, depending on their lengths and the mechanisms of load transfer to the soil: (a) point bearing piles, (b) friction piles, and (c) compaction piles. Point Bearing Piles If soil-boring records establish the presence of bedrock or rocklike material at a site within a reasonable depth, piles can be extended to the rock surface. (See Figure 11.6a.) In this case, the ultimate capacity of the piles depends entirely on the load-bearing capacity of the underlying material; thus, the piles are called point bearing piles. In most of these cases, the necessary length of the pile can be fairly well established. If, instead of bedrock, a fairly compact and hard stratum of soil is encountered at a reasonable depth, piles can be extended a few meters into the hard stratum. (See Figure 11.6b.) Piles with pedestals can be constructed on the bed of the hard stratum, and the ultimate pile load may be expressed as Qu 5 Qp 1 Qs (11.5) where Qp 5 load carried at the pile point Qs 5 load carried by skin friction developed at the side of the pile (caused by shearing resistance between the soil and the pile) If Qs is very small, Qs < Qp (11.6) In this case, the required pile length may be estimated accurately if proper subsoil exploration records are available. 11.3 Estimating Pile Length Qu Qu Weak soil L Qu Qs L Qs Weak soil Lb Qp Rock Qu Qp (a) 547 L Weak soil Strong soil layer Qp Qu Q p Qp Qu Qs Lb depth of penetration into bearing stratum (b) (c) Figure 11.6 (a) and (b) Point bearing piles; (c) friction piles Friction Piles When no layer of rock or rocklike material is present at a reasonable depth at a site, point bearing piles become very long and uneconomical. In this type of subsoil, piles are driven through the softer material to specified depths. (See Figure 11.6c.) The ultimate load of the piles may be expressed by Eq. (11.5). However, if the value of Qp is relatively small, then Qu < Qs (11.7) These piles are called friction piles, because most of their resistance is derived from skin friction. However, the term friction pile, although used often in the literature, is a misnomer: In clayey soils, the resistance to applied load is also caused by adhesion. The lengths of friction piles depend on the shear strength of the soil, the applied load, and the pile size. To determine the necessary lengths of these piles, an engineer needs a good understanding of soil–pile interaction, good judgment, and experience. Theoretical procedures for calculating the load-bearing capacity of piles are presented later in the chapter. Compaction Piles Under certain circumstances, piles are driven in granular soils to achieve proper compaction of soil close to the ground surface. These piles are called compaction piles. The lengths of compaction piles depend on factors such as (a) the relative density of the soil before compaction, (b) the desired relative density of the soil after compaction, and (c) the required depth of compaction. These piles are generally short; however, some field tests are necessary to determine a reasonable length. 548 Chapter 11: Pile Foundations 11.4 Installation of Piles Most piles are driven into the ground by means of hammers or vibratory drivers. In special circumstances, piles can also be inserted by jetting or partial augering. The types of hammer used for pile driving include (a) the drop hammer, (b) the single-acting air or steam hammer, (c) the double-acting and differential air or steam hammer, and (d) the diesel hammer. In the driving operation, a cap is attached to the top of the pile. A cushion may be used between the pile and the cap. The cushion has the effect of reducing the impact force and spreading it over a longer time; however, the use of the cushion is optional. A hammer cushion is placed on the pile cap. The hammer drops on the cushion. Figure 11.7 illustrates various hammers. A drop hammer (see Figure 11.7a) is raised by a winch and allowed to drop from a certain height H. It is the oldest type of hammer used for pile driving. The main disadvantage of the drop hammer is its slow rate of blows. The principle of the single-acting air or steam hammer is shown in Figure 11.7b. The striking part, or ram, is raised by air or steam pressure and then drops by gravity. Figure 11.7c shows the operation of the double-acting and differential air or steam hammer. Air or steam is used both to raise the ram and to push it downward, thereby increasing the impact velocity of the ram. The diesel hammer (see Figure 11.7d) consists essentially of a ram, an anvil block, and a fuel-injection system. First the ram is raised and fuel is injected near the anvil. Then the ram is released. When the ram drops, it compresses the air–fuel mixture, which ignites. This action, in effect, pushes the pile downward and raises the ram. Diesel hammers work well under hard driving conditions. In soft soils, the downward movement of the pile is rather large, and the upward movement of the ram is small. This differential may Exhaust Cylinder Intake Ram Ram Hammer cushion Hammer cushion Pile cap Pile cap Pile cushion Pile cushion Pile Pile (a) (b) Figure 11.7 Pile-driving equipment: (a) drop hammer; (b) single-acting air or steam hammer 11.4 Installation of Piles 549 Exhaust and Intake Cylinder Exhaust and Intake Ram Ram Anvil Hammer cushion Pile cushion Pile cap Pile cap Pile cushion Hammer cushion Pile Pile (d) (c) Static weight Oscillator Clamp Pile (e) (f) Figure 11.7 (continued) Pile-driving equipment: (c) double-acting and differential air or steam hammer; (d) diesel hammer; (e) vibratory pile driver; (f) photograph of a vibratory pile driver (Courtesy of Michael W. O’Neill, University of Houston) 550 Chapter 11: Pile Foundations Table 11.4 Examples of Commercially Available Pile-Driving Hammers Maker of hammer† Model No. Rated energy kN-m Blows/min Ram weight kN V M M M R R 400C S-20 S-8 S-5 5/O 2/O Single acting 153.9 81.3 35.3 22.0 77.1 44.1 100 60 53 60 44 50 177.9 89.0 35.6 22.2 77.8 44.5 V V V V R 200C 140C 80C 65C 150C Double acting or differential 68.1 48.8 33.1 26.0 66.1 98 103 111 117 95–105 89.0 62.3 35.6 28.9 66.7 V V M M 4N100 IN100 DE40 DE30 Diesel 58.8 33.4 43.4 30.4 50–60 50–60 48 48 23.5 13.3 17.8 12.5 Hammer type † V—Vulcan Iron Works, Florida M—McKiernan-Terry, New Jersey R—Raymond International, Inc., Texas not be sufficient to ignite the air–fuel system, so the ram may have to be lifted manually. Table 11.4 provides some examples of commercially available pile-driving hammers. The principles of operation of a vibratory pile driver are shown in Figure 11.7e. This driver consists essentially of two counterrotating weights. The horizontal components of the centrifugal force generated as a result of rotating masses cancel each other. As a result, a sinusoidal dynamic vertical force is produced on the pile and helps drive the pile downward. Figure 11.7f is a photograph of a vibratory pile driver. Figure 11.8 shows a pile-driving operation in the field. Jetting is a technique that is sometimes used in pile driving when the pile needs to penetrate a thin layer of hard soil (such as sand and gravel) overlying a layer of softer soil. In this technique, water is discharged at the pile point by means of a pipe 50 to 75 mm in diameter to wash and loosen the sand and gravel. Piles driven at an angle to the vertical, typically 14 to 20°, are referred to as batter piles. Batter piles are used in group piles when higher lateral load-bearing capacity is required. Piles also may be advanced by partial augering, with power augers (see Chapter 2) used to predrill holes part of the way. The piles can then be inserted into the holes and driven to the desired depth. Piles may be divided into two categories based on the nature of their placement: displacement piles and nondisplacement piles. Driven piles are displacement piles, because they move some soil laterally; hence, there is a tendency for densification of soil surrounding them. Concrete piles and closed-ended pipe piles are high-displacement piles. However, steel H-piles displace less soil laterally during driving, so they are lowdisplacement piles. In contrast, bored piles are nondisplacement piles because their placement causes very little change in the state of stress in the soil. 11.5 Load Transfer Mechanism 551 Figure 11.8 A pile-driving operation in the field (Courtesy of E. C. Shin, University of Incheon, Korea) 11.5 Load Transfer Mechanism The load transfer mechanism from a pile to the soil is complicated. To understand it, consider a pile of length L, as shown in Figure 11.9a. The load on the pile is gradually increased from zero to Q(z50) at the ground surface. Part of this load will be resisted by 552 Chapter 11: Pile Foundations Qu Q(z 0) Q(z 0) Q z z Q(z) Q1 L Q(z) 1 2 Q2 Q2 Qp Qs (a) (b) Unit frictional resistance z0 f(z) Qu Q(z) p z Qs L Pile tip Zone II Zone I Zone II zL Qp (c) (d) (e) Figure 11.9 Load transfer mechanism for piles the side friction developed along the shaft, Q1 , and part by the soil below the tip of the pile, Q2 . Now, how are Q1 and Q2 related to the total load? If measurements are made to obtain the load carried by the pile shaft, Q(z) , at any depth z, the nature of the variation found will be like that shown in curve 1 of Figure 11.9b. The frictional resistance per unit area at any depth z may be determined as f(z) 5 DQ(z) (p) ( Dz) (11.8) 11.5 Load Transfer Mechanism 553 where p 5 perimeter of the cross section of the pile. Figure 11.9c shows the variation of f(z) with depth. If the load Q at the ground surface is gradually increased, maximum frictional resistance along the pile shaft will be fully mobilized when the relative displacement between the soil and the pile is about 5 to 10 mm, irrespective of the pile size and length L. However, the maximum point resistance Q2 5 Qp will not be mobilized until the tip of the pile has moved about 10 to 25% of the pile width (or diameter). (The lower limit applies to driven piles and the upper limit to bored piles). At ultimate load (Figure 11.9d and curve 2 in Figure 11.9b), Q(z50) 5 Qu . Thus, Q1 5 Qs and Q2 5 Qp The preceding explanation indicates that Qs (or the unit skin friction, f, along the pile shaft) is developed at a much smaller pile displacement compared with the point resistance, Qp . In order to demonstrate this point, let us consider the results of a pile load test conducted in the field by Mansur and Hunter (1970). The details of the pile and subsoil conditions are as follow: Type of pile: Steel pile with 406 mm outside diameter with 8.15 mm wall thickness Type of subsoil: Sand Length of pile embedment: 16.8 m Figure 11.10a shows the load test results, which is a plot of load at the top of the pile 3Q(z50) 4 versus settlement(s). Figure 11.10b shows the plot of the load carried by the pile shaft 3Q(z) 4 at any depth. It was reported by Mansur and Hunter (1970) that, for this test, at failure Qu < 1601 kN Qp < 416 kN and Qs < 1185 kN Now, let us consider the load distribution in Figure 11.10b when the pile settlement(s) is about 2.5 mm. For this condition, Q(z50) < 667 kN Q2 < 93 kN Q1 < 574 kN Hence, at s 5 2.5 mm, Q2 93 5 (100) 5 22.4% Qp 416 and Q1 574 5 (100) 5 48.4% Qs 1185 Thus, it is obvious that the skin friction is mobilized faster at low settlement levels as compared to the point load. 554 Chapter 11: Pile Foundations Load at the top of pile, Q(z = 0) (kN) 0 400 800 1200 1600 Q(z = 0) (kN) 2000 2200 0 400 800 1200 1600 2000 0 5 3 s = 2.5 mm s = 5 mm 6 Depth, z (m) Settlement, s (mm) s = 11 mm 10 15 9 20 12 25 15 30 16.8 35 (a) (b) Figure 11.10 Load test results on a pipe pile in sand (Based on Mansur and Hunter, 1970) At ultimate load, the failure surface in the soil at the pile tip (a bearing capacity failure caused by Qp) is like that shown in Figure 11.9e. Note that pile foundations are deep foundations and that the soil fails mostly in a punching mode, as illustrated previously in Figures 3.1c and 3.3. That is, a triangular zone, I, is developed at the pile tip, which is pushed downward without producing any other visible slip surface. In dense sands and stiff clayey soils, a radial shear zone, II, may partially develop. Hence, the load displacement curves of piles will resemble those shown in Figure 3.1c. 11.6 Equations for Estimating Pile Capacity The ultimate load-carrying capacity Qu of a pile is given by the equation Qu 5 Qp 1 Qs (11.9) where Qp 5 load-carrying capacity of the pile point Qs 5 frictional resistance (skin friction) derived from the soil–pile interface (see Figure 11.11) Numerous published studies cover the determination of the values of Qp and Qs . Excellent reviews of many of these investigations have been provided by Vesic (1977), Meyerhof (1976), and Coyle and Castello (1981). These studies afford an insight into the problem of determining the ultimate pile capacity. 11.6 Equations for Estimating Pile Capacity 555 Qu Steel Soil plug Qs D (b) Open-Ended Pipe Pile Section L Lb D Steel q Soil plug d1 Qp L length of embedment Lb length of embedment in bearing stratum (a) d2 (c) H-Pile Section (Note: Ap area of steel soil plug) Figure 11.11 Ultimate load-carrying capacity of pile Point Bearing Capacity, Qp The ultimate bearing capacity of shallow foundations was discussed in Chapter 3. According to Terzaghi’s equations, qu 5 1.3crNc 1 qNq 1 0.4gBNg (for shallow square foundations) qu 5 1.3crNc 1 qNq 1 0.3gBNg (for shallow circular foundations) and Similarly, the general bearing capacity equation for shallow foundations was given in Chapter 3 (for vertical loading) as qu 5 crNcFcsFcd 1 qNqFqsFqd 1 12gBNgFgsFgd Hence, in general, the ultimate load-bearing capacity may be expressed as qu 5 crN *c 1 qN *q 1 gBN *g (11.10) where N c* , N q* , and N g* are the bearing capacity factors that include the necessary shape and depth factors. Pile foundations are deep. However, the ultimate resistance per unit area developed at the pile tip, qp , may be expressed by an equation similar in form to Eq. (11.10), although the values of N c* , N q* , and N g* will change. The notation used in this chapter for the width of a pile is D. Hence, substituting D for B in Eq. (11.10) gives qu 5 qp 5 crN c* 1 qN q* 1 gDN g* (11.11) 556 Chapter 11: Pile Foundations Because the width D of a pile is relatively small, the term gDN g* may be dropped from the right side of the preceding equation without introducing a serious error; thus, we have qp 5 crN *c 1 qrN *q (11.12) Note that the term q has been replaced by qr in Eq. (11.12), to signify effective vertical stress. Thus, the point bearing of piles is Qp 5 Ap qp 5 Ap (crN *c 1 qrN *q ) (11.13) where A p 5 area of pile tip cr 5 cohesion of the soil supporting the pile tip qp 5 unit point resistance qr 5 effective vertical stress at the level of the pile tip N c* , N q* 5 the bearing capacity factors Frictional Resistance, Qs The frictional, or skin, resistance of a pile may be written as Qs 5 S p DLf (11.14) where p 5 perimeter of the pile section DL 5 incremental pile length over which p and f are taken to be constant f 5 unit friction resistance at any depth z The various methods for estimating Qp and Qs are discussed in the next several sections. It needs to be reemphasized that, in the field, for full mobilization of the point resistance (Qp ), the pile tip must go through a displacement of 10 to 25% of the pile width (or diameter). Allowable Load, Qall After the total ultimate load-carrying capacity of a pile has been determined by summing the point bearing capacity and the frictional (or skin) resistance, a reasonable factor of safety should be used to obtain the total allowable load for each pile, or Qall 5 Qu FS where Qall 5 allowable load-carrying capacity for each pile FS 5 factor of safety The factor of safety generally used ranges from 2.5 to 4, depending on the uncertainties surrounding the calculation of ultimate load. 11.7 Meyerhof’s Method for Estimating Qp 11.7 557 Meyerhof’s Method for Estimating Qp Sand The point bearing capacity, qp , of a pile in sand generally increases with the depth of embedment in the bearing stratum and reaches a maximum value at an embedment ratio of Lb>D 5 (Lb>D) cr . Note that in a homogeneous soil Lb is equal to the actual embedment length of the pile, L. However, where a pile has penetrated into a bearing stratum, Lb , L. Beyond the critical embedment ratio, (Lb>D) cr , the value of qp remains constant (qp 5 ql ). That is, as shown in Figure 11.12 for the case of a homogeneous soil, L 5 Lb . For piles in sand, cr 5 0, and Eq. (11.13) simpifies to Qp 5 Ap qp 5 Ap qrN *q (11.15) The variation of N *q with soil friction angle fr is shown in Figure 11.13. The interpolated values of N *q for various friction angles are also given in Table 11.5. However, Qp should not exceed the limiting value A pql ; that is, Qp 5 A pqrN *q < A pql Unit point resistance, qp 1000 800 600 400 200 100 80 60 N*q Figure 11.13 Variation of the maximum values of N *q with soil friction angle fr (From Meyerhof, G. G. (1976). “Bearing Capacity and Settlement of Pile Foundations,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 102, No. GT3, pp. 197–228. With permission from ASCE.) (11.16) 40 20 (Lb /D)cr N*q 10 8 6 4 qp ql L/D Lb /D Figure 11.12 Nature of variation of unit point resistance in a homogeneous sand 2 1 0 10 20 30 Soil friction angle, (deg) 40 45 558 Chapter 11: Pile Foundations Table 11.5 Interpolated Values of N *q Based on Meyerhof’s Theory Soil friction angle, f (deg) Nq* 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 12.4 13.8 15.5 17.9 21.4 26.0 29.5 34.0 39.7 46.5 56.7 68.2 81.0 96.0 115.0 143.0 168.0 194.0 231.0 276.0 346.0 420.0 525.0 650.0 780.0 930.0 The limiting point resistance is ql 5 0.5 paN *q tan fr (11.17) where pa 5 atmospheric pressure (5100 kN>m2) fr 5 effective soil friction angle of the bearing stratum A good example of the concept of the critical embedment ratio can be found from the field load tests on a pile in sand at the Ogeechee River site reported by Vesic (1970). The pile tested was a steel pile with a diameter of 457 mm. Table 11.6 shows the ultimate resistance at various depths. Figure 11.14 shows the plot of qp with depth obtained from the field tests along with the range of standard penetration resistance at the site. From the figure, the following observations can be made. 1. There is a limiting value of qp. For the tests under consideration, it is about 12,000 kN/m2. 2. The (L>D)cr value is about 16 to 18. 11.7 Meyerhof’s Method for Estimating Qp 559 Table 11.6 Ultimate Point Resistance, qp, of Test Pile at the Ogeechee River Site As reported by Vesic (1970) Pile diameter, D (m) Depth of embedment, L (m) L ,D qp (kN , m2) 0.457 0.457 0.457 0.457 0.457 3.02 6.12 8.87 12.0 15.00 6.61 13.39 19.4 26.26 32.82 3,304 9,365 11,472 11,587 13,971 0 Pile point resistance, qp(kN/m2) 4000 8000 12,000 16,000 20,000 0 2 Pile point resistance 4 Depth (m) 6 Range of N60 at site 8 10 12 14 0 10 20 30 N60 40 50 Figure 11.14 Vesic’s pile test (1970) result—variation of qp and N60 with depth 3. The average N60 value is about 30 for L>D > (L>D) cr. Using Eq. (11.37), the limiting point resistance is 4pa N60 5 (4)(100)(30) 5 12,000 kN/m2. This value is generally consistent with the field observation. Clay (f 5 0) For piles in saturated clays under undrained conditions (f 5 0), the net ultimate load can be given as Qp < N *c cuA p 5 9cuA p where cu 5 undrained cohesion of the soil below the tip of the pile. (11.18) 560 Chapter 11: Pile Foundations 11.8 Vesic’s Method for Estimating Q p Sand Vesic (1977) proposed a method for estimating the pile point bearing capacity based on the theory of expansion of cavities. According to this theory, on the basis of effective stress parameters, we may write Qp 5 A pqp 5 A p sor N *s (11.19) where sor 5 mean effective normal ground stress at the level of the pile point 1 1 2Ko ≤qr 3 Ko 5 earth pressure coefficient at rest 5 1 2 sin fr 5¢ (11.20) (11.21) and N *s 5 bearing capacity factor Note that Eq. (11.19) is a modification of Eq. (11.15) with N s* 5 3N *q (1 1 2Ko ) (11.22) According to Vesic’s theory, N s* 5 f(Irr ) (11.23) where Irr 5 reduced rigidity index for the soil. However, Irr 5 Ir 1 1 Ir D (11.24) where Ir 5 rigidity index 5 Es Gs 5 2(1 1 ms ) qr tan fr qr tan fr (11.25) Es 5 modulus of elasticity of soil ms 5 Poisson’s ratio of soil Gs 5 shear modulus of soil D 5 average volumatic strain in the plastic zone below the pile point The general ranges of Ir for various soils are Sand(relative density 5 50% to 80%): 75 to 150 Silt : 50 to 75 In order to estimate Ir [Eq. (11.25)] and hence Irr [Eq. (11.24)], the following approximations may be used (Chen and Kulhawy, 1994) 11.8 Vesic’s Method for Estimating Qp Es 5m pa 561 (11.26) where pa 5 atmospheric pressure ( < 100 kN>m2 ) 100 to 200(loose soil) m 5 c200 to 500(medium dense soil) 500 to 1000(dense soil) ms 5 0.1 1 0.3a fr 2 25 b (for 25° # f r # 45°) 20 (11.27) fr 2 25 qr b pa 20 (11.28) D 5 0.005a1 2 On the basis of cone penetration tests in the field, Baldi et al. (1981) gave the following correlations for Ir : Ir 5 and 300 Fr (%) Ir 5 170 Fr (%) (for mechanical cone penetration) (11.29) (for electric cone penetration) (11.30) For the definition of Fr , see Eq. (2.41). Table 11.7 gives the values of N s* for various values of Irr and fr. Clay (f 5 0) In saturated clay (f 5 0 condition), the net ultimate point bearing capacity of a pile can be approximated as Qp 5 A pqp 5 A pcuN c* (11.31) where cu 5 undrained cohesion According to the expansion of cavity theory of Vesic (1977), N *c 5 4 p (ln Irr 1 1) 1 1 1 3 2 (11.32) The variations of N c* with Irr for f 5 0 condition are given in Table 11.8. Now, referring to Eq. (11.24) for saturated clay with no volume change, D 5 0. Hence, Irr 5 Ir (11.33) 562 12.12 13.18 14.33 15.57 16.90 18.24 19.88 21.55 23.34 25.28 27.36 29.60 32.02 34.63 37.44 40.47 43.74 47.27 51.08 55.20 59.66 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 15.95 17.47 19.12 20.91 22.85 24.95 27.22 29.68 32.34 35.21 38.32 41.68 45.31 49.24 53.50 58.10 63.07 68.46 74.30 80.62 87.48 20 20.98 23.15 25.52 28.10 30.90 33.95 37.27 40.88 44.80 49.05 53.67 58.68 64.13 70.03 76.45 83.40 90.96 99.16 108.08 117.76 128.28 40 24.64 27.30 30.21 33.40 36.87 40.66 44.79 49.30 54.20 59.54 65.36 71.69 78.57 86.05 94.20 103.05 112.68 123.16 134.56 146.97 160.48 60 27.61 30.69 34.06 37.75 41.79 46.21 51.03 56.30 62.05 68.33 75.17 82.62 90.75 99.60 109.24 119.74 131.18 143.64 157.21 172.00 188.12 80 30.16 33.60 37.37 41.51 46.05 51.02 56.46 62.41 68.92 76.02 83.78 92.24 101.48 111.56 122.54 134.52 147.59 161.83 177.36 194.31 212.79 100 39.70 44.53 49.88 55.77 62.27 69.43 77.31 85.96 95.46 105.90 117.33 129.87 143.61 158.65 175.11 193.13 212.84 234.40 257.99 283.80 312.03 200 46.61 52.51 59.05 66.29 74.30 83.14 92.90 103.66 115.51 128.55 142.89 158.65 175.95 194.94 215.78 238.62 263.67 291.13 321.22 354.20 390.35 300 52.24 59.02 66.56 74.93 84.21 94.48 105.84 118.39 132.24 147.51 164.33 182.85 203.23 225.62 250.23 277.26 306.94 339.52 375.28 414.51 457.57 400 57.06 64.62 73.04 82.40 92.80 104.33 117.11 131.24 146.87 164.12 183.16 204.14 227.26 252.71 280.71 311.50 345.34 382.53 423.39 468.28 517.58 500 From “Design of Pile Foundations,” by A. S. Vesic. SYNTHESIS OF HIGHWAY PRACTICE by AMERICAN ASSOCIATION OF STATE HIGHWAY AND TRANSPORT. Copyright 1969 by TRANSPORTATION RESEARCH BOARD. Reproduced with permission of TRANSPORTATION RESEARCH BOARD in the format Textbook via Copyright Clearance Center. 10 f9 I rr Table 11.7 Bearing Capacity Factors N s* Based on the Theory of Expansion of Cavities 11.9 Coyle and Castello’s Method for Estimating Qp in Sand 563 Table 11.8 Variation of N c* with Irr for f 5 0 Condition based on Vesic’s Theory Irr Nc* 10 20 40 60 80 100 200 300 400 500 6.97 7.90 8.82 9.36 9.75 10.04 10.97 11.51 11.89 12.19 For f 5 0, Ir 5 Es 3cu (11.34) O’ Neill and Reese (1999) suggested the following approximate relationships for Ir and the undrained cohesion, cu. cu pa Ir 0.24 0.48 ⱖ0.96 50 150 250⫺300 Note: pa ⫽ atmospheric pressure < 100 kN>m2. The preceding values can be approximated as Ir 5 347a 11.9 cu b 2 33 # 300 pa (11.35) Coyle and Castello’s Method for Estimating Qp in Sand Coyle and Castello (1981) analyzed 24 large-scale field load tests of driven piles in sand. On the basis of the test results, they suggested that, in sand, Qp 5 qrN q*A p where qr 5 effective vertical stress at the pile tip N *q 5 bearing capacity factor Figure 11.15 shows the variation of N *q with L>D and the soil friction angle fr. (11.36) 564 Chapter 11: Pile Foundations 10 Bearing capacity factor, N*q 20 40 60 80 100 200 0 10 Embedment ratio, L/D 20 30 40 50 60 32° 36° 40° 30° 34° 38° Figure 11.15 Variation of N q* with L>D (Redrawn after Coyle and Costello, 1981) 70 Example 11.1 Consider a 15-m long concrete pile with a cross section of 0.45 m 3 0.45 m fully embedded in sand. For the sand, given: unit weight, g 5 17kN>m3; and soil friction angle, f r 5 35°. Estimate the ultimate point Qp with each of the following: a. b. c. d. Meyerhof’s method Vesic’s method The method of Coyle and Castello based on the results of parts a, b, and c, adopt a value for Qp Solution Part a From Eqs. (11.16) and (11.17), Qp 5 A pq rN q* # A p (0.5paN q* tan fr ) For fr 5 35°, the value of N q* < 143 (Table 11.5). Also, q r 5 gL 5 (17) (15) 5 255kN>m2. Thus, A pq rN *q 5 (0.45 3 0.45) (255) (143) < 7384kN Again, A p (0.5paN *q tan fr ) 5 (0.45 3 0.45) 3(0.5) (100) (143) ( tan 35)4 < 1014 kN Hence, Qp 5 1014 kN. Part b From Eq. (11.19), 11.9 Coyle and Castello’s Method for Estimating Qp in Sand 565 Qp 5 A psor N s* sor 5 c 1 1 2(1 2 sin fr ) 1 1 2(1 2 sin 35) d qr 5 a b (17 3 15) 3 3 5 139.96kN>m2 From Eq. (11.26), Es 5m pa Assume m < 250 (medium sand). So, Es 5 (250) (100) 5 25,000 kN>m2 From Eq. (11.27), ms 5 0.1 1 0.3a fr 2 25 35 2 25 b 5 0.1 1 0.3a b 5 0.25 20 20 From Eq. (11.28), D 5 0.005a1 2 fr 2 25 qr 35 2 25 17 3 15 ba b 5 0.005a1 2 ba b 5 0.0064 pa 20 20 100 From Eq. (11.25), Ir 5 Es 25,000 5 5 56 2(1 1 ms )q r tan fr (2) (1 1 0.25) (17 3 15) ( tan 35) From Eq. (11.24), Irr 5 Ir 56 5 5 41.2 1 1 Ir D 1 1 (56) (0.0064) From Table 11.7, for fr 5 35° and Irr 5 41.2, the value of N *s < 55. Hence, Qp 5 A psor N s* 5 (0.45 3 0.45) (139.96) (55) < 1559 kN Part c From Eq. (11.36), Qp 5 q rN *q A p L 15 5 5 33.3 D 0.45 For fr 5 35° and L>D 5 33.3, the value of N *q is about 48 (Figure 11.15). Thus, Qp 5 q rN *q A p 5 (15 3 17) (48) (0.45 3 0.45) < 2479 kN Part d It appears that Qp obtained from the method of Coyle and Castello is too large. Thus, the average of the results from parts a and b is 566 Chapter 11: Pile Foundations 1014 1 1559 5 1286.5 kN 2 Use Qp 5 1250 kN. ■ Example 11.2 Consider a pipe pile (flat driving point—see Figure 11.2d) having an outside diameter of 406 mm. The embedded length of the pile in layered saturated clay is 30 m. The following are the details of the subsoil: Depth from ground surface (m) Saturated unit weight, g(kN>m3 ) cu (kN>m2 ) 0–5 5–10 10–30 18 18 19.6 30 30 100 The groundwater table is located at a depth of 5 m from the ground surface. Estimate Qp by using a. Meyerhof’s method b. Vesic’s method Solution Part a From Eq. (11.18), Qp 5 9cuA p The tip of the pile is resting on a clay with cu 5 100 kN>m2. So, p 406 2 Qp 5 (9) (100) c a b a b d 5 116.5 kN 4 1000 Part b From Eq. (11.31), Qp 5 A pcuN *c From Eq. (11.35), Ir 5 Irr 5 347a cu 100 b 2 33 5 347a b 2 33 5 314 pa 100 So we use Irr 5 300. From Table 11.8 for Irr 5 300, the value of N c* 5 11.51. Thus, p 406 2 Qp 5 A pcuN *c 5 c a b a b d (100) (11.51) 5 149.0 kN 4 1000 11.10 Correlations for Calculating Qp with SPT and CPT Results 567 Note: The average value of Qp is 116.5 1 149.0 < 133 kN 2 11.10 ■ Correlations for Calculating Qp with SPT and CPT Results On the basis of field observations, Meyerhof (1976) also suggested that the ultimate point resistance qp in a homogeneous granular soil (L 5 Lb ) may be obtained from standard penetration numbers as L qp 5 0.4paN60 # 4paN60 (11.37) D where N60 5 the average value of the standard penetration number near the pile point (about 10D above and 4D below the pile point) pa 5 atmospheric pressure ( < 100 kN>m2 or 2000 lb>ft 2 ) Briaud et al. (1985) suggested the following correlation for qp in granular soil with the standard penetration resistance N60. qp 5 19.7pa (N60 ) 0.36 Meyerhof (1956) also suggested that qp < qc (in granular soil) where qc 5 cone penetration resistance. (11.38) (11.39) Example 11.3 Consider a concrete pile that is 0.305 m 3 0.305 m in cross section in sand. The pile is 15.2 m long. The following are the variations of N60 with depth. Depth below ground surface (m) 1.5 3.0 4.5 6.0 7.5 9.0 10.5 12.0 13.5 15.0 16.5 18.0 19.5 21.0 N 60 8 10 9 12 14 18 11 17 20 28 29 32 30 27 568 Chapter 11: Pile Foundations a. Estimate Qp using Eq. (11.37). b. Estimate Qp using Eq. (11.38). Solution Part a The tip of the pile is 15.2 m below the ground surface. For the pile, D 5 0.305 m. The average of N60 10D above and about 5D below the pile tip is N60 5 17 1 20 1 28 1 29 5 23.5 < 24 4 From Eq. (11.37) Qp 5 A p (qp ) 5 A p c0.4paN60 a A p c0.4paN60 a L b d # A p (4paN60 ) D L 15.2 b d 5 (0.305 3 0.305) c (0.4) (100) (24) a b d 5 4450.6 kN D 0.305 A p (4paN60 ) 5 (0.305 3 0.305) 3(4) (100) (24)4 5 893 kN Thus, Qp 5 893 kN Part b From Eq. (11.38), Qp 5 A pqp 5 A p 319.7pa (N60 ) 0.364 5 (0.305 3 0.305) 3(19.7) (100) (24) 0.364 5 575.4 kN 11.11 ■ Frictional Resistance (Qs) in Sand According to Eq. (11.14), the frictional resistance Qs 5 Sp DLf The unit frictional resistance, f, is hard to estimate. In making an estimation of f, several important factors must be kept in mind: 1. The nature of the pile installation. For driven piles in sand, the vibration caused during pile driving helps densify the soil around the pile. The zone of sand densification may be as much as 2.5 times the pile diameter, in the sand surrounding the pile. 2. It has been observed that the nature of variation of f in the field is approximately as shown in Figure 11.16. The unit skin friction increases with depth more or 11.11 Frictional Resistance (Qs) in Sand 569 Unit frictional resistance, f D L z f Ko L L Depth (a) (b) Figure 11.16 Unit frictional resistance for piles in sand less linearly to a depth of Lr and remains constant thereafter. The magnitude of the critical depth Lr may be 15 to 20 pile diameters. A conservative estimate would be Lr < 15D (11.40) 3. At similar depths, the unit skin friction in loose sand is higher for a highdisplacement pile, compared with a low-displacement pile. 4. At similar depths, bored, or jetted, piles will have a lower unit skin friction compared with driven piles. Taking into account the preceding factors, we can give the following approximate relationship for f (see Figure 11.16): For z 5 0 to Lr, f 5 Ksor tan d r (11.41) f 5 fz5Lr (11.42) and for z 5 Lr to L, In these equations, K 5 effective earth pressure coefficient sor 5 effective vertical stress at the depth under consideration dr 5 soil-pile friction angle In reality, the magnitude of K varies with depth; it is approximately equal to the Rankine passive earth pressure coefficient, Kp , at the top of the pile and may be less than 570 Chapter 11: Pile Foundations the at-rest pressure coefficient, Ko , at a greater depth. Based on presently available results, the following average values of K are recommended for use in Eq. (11.41): K Pile type <Ko 5 1 2 sin fr <Ko 5 1 2 sin fr to 1.4Ko 5 1.4(1 2 sin fr) <Ko 5 1 2 sin fr to 1.8Ko 5 1.8(1 2 sin fr) Bored or jetted Low-displacement driven High-displacement driven The values of dr from various investigations appear to be in the range from 0.5fr to 0.8fr. Based on load test results in the field, Mansur and Hunter (1970) reported the following average values of K. H-pilesccK 5 1.65 Steel pipe pilesccK 5 1.26 Precast concrete pilesccK 5 1.5 Coyle and Castello (1981), in conjunction with the material presented in Section 11.9, proposed that Qs 5 favpL 5 (Ksor tan dr )pL (11.43) where sor 5 average effective overburden pressure dr 5 soil–pile friction angle 5 0.8fr The lateral earth pressure coefficient K, which was determined from field observations, is shown in Figure 11.17. Thus, if that figure is used, Qs 5 Ksor tan(0.8fr)pL (11.44) Correlation with Standard Penetration Test Results Meyerhof (1976) indicated that the average unit frictional resistance, fav , for highdisplacement driven piles may be obtained from average standard penetration resistance values as fav 5 0.02pa (N60 ) (11.45) where (N60 ) 5 average value of standard penetration resistance pa 5 atmospheric pressure (<100 kN>m2 or 2000 lb>ft 2 ) For low-displacement driven piles fav 5 0.01pa (N60 ) (11.46) 11.11 Frictional Resistance (Qs) in Sand 0.15 0.2 0 Earth pressure coefficient, K 1.0 2 571 5 30° 31° 5 32° 33° 10 35° Embedment ratio, L /D 34° 36° 15 20 25 30 Figure 11.17 Variation of K with L>D (Redrawn after Coyle and Castello, 1981) 35 36 Briaud et al. (1985) suggested that fav < 0.224pa (N60 ) 0.29 (11.47) Qs 5 pLfav (11.48) Thus, Correlation with Cone Penetration Test Results Nottingham and Schmertmann (1975) and Schmertmann (1978) provided correlations for estimating Qs using the frictional resistance (fc ) obtained during cone penetration tests. According to this method f 5 arfc (11.49) The variations of ar with z>D for electric cone and mechanical cone penetrometers are shown in Figures 11.18 and 11.19, respectively. We have Qs 5 Sp(DL)f 5 Sp(DL)arfc (11.50) 572 Chapter 11: Pile Foundations 3.0 Schmertmann (1978); Nottingham and Schmertmann (1975) Steel pile 2.0 Timber pile Concrete pile 1.0 0 0 10 20 L /D 30 40 Figure 11.18 Variation of ar with embedment ratio for pile in sand: electric cone penetrometer 2.0 Schmertmann (1978); Nottingham and Schmertmann (1975) 1.5 Steel pile Timber pile 1.0 Concrete pile 0.5 0 0 10 20 L /D 30 40 Figure 11.19 Variation of ar with embedment ratio for piles in sand: mechanical cone penetrometer Example 11.4 Refer to the pile described in Example 11.3. Estimate the magnitude of Qs for the pile. a. Use Eq. (11.45). b. Use Eq. (11.47). 11.11 Frictional Resistance (Qs) in Sand 573 c. Considering the results in Example 11.3, determine the allowable load-carrying capacity of the pile based on Meyerhof’s method and Briaud’s method. Use a factor of safety, FS 5 3. Solution The average N60 value for the sand for the top 15.2 m is N60 5 8 1 10 1 9 1 12 1 14 1 18 1 11 1 17 1 20 1 28 5 14.7 < 15 10 Part a From Eq. (11.45), fav 5 0.02pa (N60 ) 5 (0.02) (100) (15) 5 30 kN>m2 Qs 5 pLfav 5 (4 3 0.305) (15.2) (30) 5 556.2 kN Part b From Eq. (11.47), fav 5 0.224pa (N60 ) 0.29 5 (0.224) (100) (15) 0.29 5 49.13 kN>m2 Qs 5 pLfav 5 (4 3 0.305) (15.2) (49.13) 5 911.1 kN Part c Qp 1 Qs 893 1 556.2 5 483 kN 3 575.4 1 911.1 Briaud’s method: Qall 5 5 5 495.5 kN FS 3 So the allowable pile capacity may be taken to be about 490 kN. Meyerhof’s method: Qall 5 FS Qp 1 Qs 5 ■ Example 11.5 Refer to Example 11.1. For the pile, estimate the frictional resistance Qs a. Based on Eqs. (11.41) and (11.42). Use K 5 1.3 and dr 5 0.8fr. b. Based on Eq. (11.44). c. Using the results of Part d of Example 11.1, estimate the allowable bearing capacity of the pile. Use FS 5 3. Solution Part a From Eq. (11.40), Lr 5 15D 5 (15) (0.45) 5 6.75m. Refer to Eq. (11.41): At z 5 0: sor 5 0 f50 At z 5 6.75 m: sor 5 (6.75) (17) 5 114.75 kN>m2 574 Chapter 11: Pile Foundations So f 5 Ksor tan d 5 (1.3) (114.75) 3tan (0.8 3 35)4 5 79.3 kN>m3 Thus, Qs 5 (fz50 1 fz56.75m ) pLr 1 fz56.75m p(L 2 Lr ) 2 0 1 79.3 5a b (4 3 0.45) (6.75) 1 (79.3) (4 3 0.45) (15 2 6.75) 2 5 481.75 1 1177.61 5 1659.36 kN < 1659 kN Part b From Eq. (11.44), Qs 5 Ksor tan (0.8fr )pL (15) (17) sor 5 5 127.5 kN>m2 2 L 15 5 5 33.3; fr 5 35° D 0.45 From Figure 11.17, K 5 0.93 Qs 5 (1.3) (127.5) tan3 (0.8 3 35)4 (4 3 0.45) (15) 5 2380 kN Part c The average value of Qs from parts a and b is Qs(average) 5 1659 1 2380 5 2019.5 < 2020 kN 2 USE 2 From part d of Example 11.1, Qp 5 1250 kN. Thus, Qall 5 Qp 1 Qs FS 5 1250 1 2020 5 1090 kN 3 ■ Example 11.6 Consider an 18-m long concrete pile (cross section: 0.305 m 3 0.305 m) fully embedded in a sand layer. For the sand layer, the following is an approximation of the cone penetration resistance qc (mechanical cone) and the frictional resistance fc with depth. Estimate the allowable load that the pile can carry. Use FS 5 3. Depth from ground surface (m) qc (kN>m2 ) fc (kN>m2 ) 0–5 5–15 15–25 3040 4560 9500 73 102 226 11.12 Frictional (Skin) Resistance in Clay 575 Solution Qu 5 Qp 1 Qs From Eq. (11.39), qp < qc At the pile tip (i.e., at a depth of 18 m), qc < 9500 kN>m2. Thus, Qp 5 A pqc 5 (0.305 3 0.305) (9500) 5 883.7 kN To determine Qs, the following table can be prepared. (Note: L>D 5 18>0.305 5 59.) Depth from ground surface (m) 0–5 5–15 15–18 ⌬L (m) fc (kN/m2) a9 (Figure 11.19) p⌬L␣9fc (kN) 5 10 3 73 102 226 0.44 0.44 0.44 195.9 547.5 363.95 Qs 5 1107.35 kN Hence, Qu 5 Qp 1 Qs 5 883.7 1 1107.35 5 1991.05 kN Qu 1991.05 Qall 5 5 5 663.68 < 664 kN FS 3 11.12 ■ Frictional (Skin) Resistance in Clay Estimating the frictional (or skin) resistance of piles in clay is almost as difficult a task as estimating that in sand (see Section 11.11), due to the presence of several variables that cannot easily be quantified. Several methods for obtaining the unit frictional resistance of piles are described in the literature. We examine some of them next. l Method This method, proposed by Vijayvergiya and Focht (1972), is based on the assumption that the displacement of soil caused by pile driving results in a passive lateral pressure at any depth and that the average unit skin resistance is fav 5 l(sor 1 2cu ) where sor 5 mean effective vertical stress for the entire embedment length cu 5 mean undrained shear strength (f 5 0) (11.51) 576 Chapter 11: Pile Foundations Table 11.9 Variation of l with pile embedment length, L Embedment length, L (m) l 0 5 10 15 20 25 30 35 40 50 60 70 80 90 0.5 0.336 0.245 0.200 0.173 0.150 0.136 0.132 0.127 0.118 0.113 0.110 0.110 0.110 The value of l changes with the depth of penetration of the pile. (See Table 11.9.) Thus, the total frictional resistance may be calculated as Qs 5 pLfav Care should be taken in obtaining the values of sor and cu in layered soil. Figure 11.20 helps explain the reason. Figure 11.20a shows a pile penetrating three layers of clay. According to Figure 11.20b, the mean value of cu is (cu(1)L1 1 cu(2)L2 1 c )>L. Similarly, Figure 11.20c shows the plot of the variation of effective stress with depth. The mean effective stress is A1 1 A2 1 A3 1 c (11.52) sor 5 L where A 1 , A 2 , A 3 , c 5 areas of the vertical effective stress diagrams. Undrained cohesion, cu L1 L3 Area A2 cu(2) L2 L Area A1 cu(1) Area A3 cu(3) (a) Depth (b) Vertical effective stress, o Depth (c) Figure 11.20 Application of l method in layered soil 11.12 Frictional (Skin) Resistance in Clay 577 a Method According to the a method, the unit skin resistance in clayey soils can be represented by the equation f 5 acu (11.53) where a 5 empirical adhesion factor. The approximate variation of the value of a is shown in Table 11.10. It is important to realize that the values of a given in Table 11.10 may vary somewhat, since a is actually a function of vertical effective stress and the undrained cohesion. Sladen (1992) has shown that a 5 Ca sor 0.45 b cu (11.54) where sor 5 average vertical effective stress C < 0.4 to 0.5 for bored piles and $ 0.5 for driven piles The ultimate side resistance can thus be given as Qs 5 Sfp DL 5 S acup DL Table 11.10 Variation of a (interpolated values based on Terzaghi, Peck and Mesri, 1996) cu pa # 0.1 0.2 0.3 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.4 2.8 a 1.00 0.92 0.82 0.74 0.62 0.54 0.48 0.42 0.40 0.38 0.36 0.35 0.34 0.34 Note: pa 5 atmospheric pressure < 100 kN>m2 (11.55) 578 Chapter 11: Pile Foundations b Method When piles are driven into saturated clays, the pore water pressure in the soil around the piles increases. The excess pore water pressure in normally consolidated clays may be four to six times cu . However, within a month or so, this pressure gradually dissipates. Hence, the unit frictional resistance for the pile can be determined on the basis of the effective stress parameters of the clay in a remolded state (cr 5 0). Thus, at any depth, f 5 bsor (11.56) where sor 5 vertical effective stress b 5 K tan fRr fRr 5 drained friction angle of remolded clay K 5 earth pressure coefficient (11.57) Conservatively, the magnitude of K is the earth pressure coefficient at rest, or K 5 1 2 sin fRr (for normally consolidated clays) (11.58) and K 5 (1 2 sin fRr )"OCR (for overconsolidated clays) (11.59) where OCR 5 overconsolidation ratio. Combining Eqs. (11.56), (11.57), (11.58), and (11.59), for normally consolidated clays yields f 5 (1 2 sin fRr ) tan fRr sor (11.60) and for overconsolidated clays, f 5 (1 2 sin fRr )tan fRr "OCR sor (11.61) With the value of f determined, the total frictional resistance may be evaluated as Qs 5 Sfp DL Correlation with Cone Penetration Test Results Nottingham and Schmertmann (1975) and Schmertmann (1978) found the correlation for unit skin friction in clay (with f 5 0) to be f 5 arfc (11.62) The variation of ar with the frictional resistance fc is shown in Figure 11.21. Thus, Qs 5 Sfp(DL) 5 Sarfcp(DL) (11.63) 11.13 Point Bearing Capacity of Piles Resting on Rock 579 1.5 Nottingham and Schmertmann (1975); Schmertmann (1978) 1.25 1.0 0.75 Concrete and timber piles 0.5 0.25 Steel piles 0 0 0.5 1.0 fc pa 1.5 2.0 Figure 11.21 Variation of ar with fc>pa for piles in clay (pa 5 atmospheric pressure <100 kN>m2) 11.13 Point Bearing Capacity of Piles Resting on Rock Sometimes piles are driven to an underlying layer of rock. In such cases, the engineer must evaluate the bearing capacity of the rock. The ultimate unit point resistance in rock (Goodman, 1980) is approximately qp 5 qu (Nf 1 1) (11.64) where Nf 5 tan2 (45 1 fr>2) qu 5 unconfined compression strength of rock fr 5 drained angle of friction The unconfined compression strength of rock can be determined by laboratory tests on rock specimens collected during field investigation. However, extreme caution should be used in obtaining the proper value of qu , because laboratory specimens usually are small in diameter. As the diameter of the specimen increases, the unconfined compression strength decreases—a phenomenon referred to as the scale effect. For specimens larger than about 1 m in diameter, the value of qu remains approximately constant. There appears to be a fourfold to fivefold reduction of the magnitude of qu in this process. The scale effect in rock is caused primarily by randomly distributed large and 580 Chapter 11: Pile Foundations small fractures and also by progressive ruptures along the slip lines. Hence, we always recommend that qu(design) 5 qu(lab) 5 (11.65) Table 11.11 lists some representative values of (laboratory) unconfined compression strengths of rock. Representative values of the rock friction angle fr are given in Table 11.12. A factor of safety of at least 3 should be used to determine the allowable point bearing capacity of piles. Thus, Qp(all) 5 3qu(design) (Nf 1 1)4A p FS (11.66) Table 11.11 Typical Unconfined Compressive Strength of Rocks Type of rock qu MN , m2 Sandstone Limestone Shale Granite Marble 70–140 105–210 35–70 140–210 60–70 Table 11.12 Typical Values of Angle of Friction fr of Rocks Type of rock Sandstone Limestone Shale Granite Marble Angle of friction, f9 (deg) 27–45 30–40 10–20 40–50 25–30 Example 11.7 Refer to the pile in saturated clay shown in Figure 11.22. For the pile, a. Calculate the skin resistance (Qs ) by (1) the a method, (2) the l method, and (3) the b method. For the b method, use fRr 5 30° for all clay layers. The top 10 m of clay is normally consolidated. The bottom clay layer has an OCR 5 2. (Note: diameter of pile ⫽ 406 mm) b. Using the results of Example 11.2, estimate the allowable pile capacity (Qall ) . Use FS 5 4. 11.13 Point Bearing Capacity of Piles Resting on Rock 581 o (kN/m2) Saturated clay cu(1) 30 kN/m2 18 kN/m3 5m A1 225 Groundwater table cu(1) 30 kN/m2 Clay 18 kN/m3 5m 5m 90 A2 552.38 130.95 10 m z Clay cu(2) 100 kN/m2 sat 19.6 kN/m3 20 m A3 4577 30 m (a) 326.75 (b) Figure 11.22 Estimation of the load bearing capacity of a driven-pipe pile Solution Part a (1) From Eq. (11.55), Qs 5 SacupDL [Note: p 5 p(0.406) 5 1.275m] Now the following table can be prepared. Depth (m) DL (m) cu (kN/m2) a (Table 11.10) ␣cup DL (kN) 0–5 5–10 10–30 5 5 20 30 30 100 0.82 0.82 0.48 156.83 156.83 1224.0 Qs < 1538 kN (2) From Eq. 11.51, fav 5 l(sor 1 2cu ). Now, the average value of cu is cu(1) (10) 1 cu(2) (20) 30 5 (30) (10) 1 (100) (20) 5 76.7 kN>m2 30 To obtain the average value of sor , the diagram for vertical effective stress variation with depth is plotted in Figure 11.22b. From Eq. (11.52), sor 5 A1 1 A2 1 A3 225 1 552.38 1 4577 5 5 178.48 kN>m2 L 30 582 Chapter 11: Pile Foundations From Table 11.9, the magnitude of l is 0.136. So fav 5 0.1363178.48 1 (2) (76.7) 4 5 45.14 kN>m2 Hence, Qs 5 pLfav 5 p(0.406) (30) (45.14) 5 1727 kN (3) The top layer of clay (10 m) is normally consolidated, and fRr 5 30°. For z 5 0–5 m, from Eq. (11.60), we have fav(1) 5 (1 2 sin fRr ) tan fRr sor 5 (1 2 sin 30°) (tan 30°) ¢ 0 1 90 ≤ 5 13.0 kN>m2 2 Similarly, for z 5 5–10 m. fav(2) 5 (1 2 sin 30°) (tan 30°) ¢ 90 1 130.95 ≤ 5 31.9 kN>m2 2 For z 5 10–30 m from Eq. (11.61), fav 5 (1 2 sin fRr )tan fRr "OCR sor For OCR 5 2, fav(3) 5 (1 2 sin 30°) (tan 30°)"2¢ 130.95 1 326.75 ≤ 5 93.43 kN>m2 2 So, Qs 5 p3fav(1) (5) 1 fav(2) (5) 1 fav(3) (20)4 5 (p) (0.406) 3(13) (5) 1 (31.9) (5) 1 (93.43) (20)4 5 2670 kN Part b Qu 5 Qp 1 Qs From Example 11.2, Qp < 116.5 1 149 < 133 kN 2 Again, the values of Qs from the a method and l method are close. So, Qs < Qall 1538 1 1727 5 1632.5 kN 2 Qu 133 1 1632.5 5 5 5 441.4 kN < 441 kN FS 4 ■ 11.14 Pile Load Tests 583 Example 11.8 A concrete pile 305 mm 3 305 mm in cross section is driven to a depth of 20 m below the ground surface in a saturated clay soil. A summary of the variation of frictional resistance fc obtained from a cone penetration test is as follows: Depth (m) Friction resistance, fc (kg/cm2) 0–6 6–12 12–20 0.35 0.56 0.72 Estimate the frictional resistance Qs for the pile. Solution We can prepare the following table: Depth (m) fc (kN/m2) a9 (Figure 11.21) DL (m) a9fc p(DL) [Eq. (11.63)] (kN) 0 –6 6 –12 12–20 34.34 54.94 70.63 0.84 0.71 0.63 6 6 8 211.5 285.5 434.2 (Note: p 5 (4) (0.305) 5 1.22 m) Thus, Qs 5 g arfc p(DL) 5 931 kN 11.14 ■ Pile Load Tests In most large projects, a specific number of load tests must be conducted on piles. The primary reason is the unreliability of prediction methods. The vertical and lateral loadbearing capacity of a pile can be tested in the field. Figure 11.23a shows a schematic diagram of the pile load arrangement for testing axial compression in the field. The load is applied to the pile by a hydraulic jack. Step loads are applied to the pile, and sufficient time is allowed to elapse after each load so that a small amount of settlement occurs. The settlement of the pile is measured by dial gauges. The amount of load to be applied for each step will vary, depending on local building codes. Most building codes require that each step load be about one-fourth of the proposed working load. The load test should be carried out to at least a total load of two times the proposed working load. After the desired pile load is reached, the pile is gradually unloaded. 584 Chapter 11: Pile Foundations Beam Hydraulic jack Dial gauge Reference beam Anchor pile Test pile (a) Q1 Q2 st(1) Load, Q Load, Q st(2) Qu Loading se(1) Qu se(2) Unloading 1 Settlement Net settlement, snet (b) 2 (c) Figure 11.23 (a) Schematic diagram of pile load test arrangement (b) plot of load against total settlement; (c) plot of load against net settlement Figure 11.23b shows a load–settlement diagram obtained from field loading and unloading. For any load Q, the net pile settlement can be calculated as follows: When Q 5 Q1 , Net settlement, snet(1) 5 st(1) 2 se(1) When Q 5 Q2 , Net settlement, snet(2) 5 st(2) 2 se(2) ( where snet 5 net settlement se 5 elastic settlement of the pile itself st 5 total settlement These values of Q can be plotted in a graph against the corresponding net settlement, snet , as shown in Figure 11.23c. The ultimate load of the pile can then be determined from the graph. 11.14 Pile Load Tests 585 Pile settlement may increase with load to a certain point, beyond which the load–settlement curve becomes vertical. The load corresponding to the point where the curve of Q versus snet becomes vertical is the ultimate load, Qu , for the pile; it is shown by curve 1 in Figure 11.23c. In many cases, the latter stage of the load–settlement curve is almost linear, showing a large degree of settlement for a small increment of load; this is shown by curve 2 in the figure. The ultimate load, Qu , for such a case is determined from the point of the curve of Q versus snet where this steep linear portion starts. One of the methods to obtain the ultimate load Qu from the load-settlement plot is that proposed by Davisson (1973). Davisson’s method is used more often in the field and is described here. Referring to Figure 11.24, the ultimate load occurs at a settlement level (su ) of su (mm) 5 0.012Dr 1 0.1a QuL D b1 Dr ApEp where Qu is in kN D is in mm Dr 5 reference pile diameter or width (5 300mm) L 5 pile length (mm) A p 5 area of pile cross section (mm2 ) Ep 5 Young’s modulus of pile material (kN>mm2 ) Qu Load, Q (kN) 0.12 Dr+ 0.1 D/Dr QuL AE Eq. (11.67) Settlement, s (mm) Figure 11.24 Davisson’s method for determination of Qu (11.67) 586 Chapter 11: Pile Foundations The application of this procedure is shown in Example 11.9. The load test procedure just described requires the application of step loads on the piles and the measurement of settlement and is called a load-controlled test. Another technique used for a pile load test is the constant-rate-of-penetration test, wherein the load on the pile is continuously increased to maintain a constant rate of penetration, which can vary from 0.25 to 2.5 mm>min (0.01 to 0.1 in.>min). This test gives a load–settlement plot similar to that obtained from the load-controlled test. Another type of pile load test is cyclic loading, in which an incremental load is repeatedly applied and removed. In order to conduct a load test on piles, it is important to take into account the time lapse after the end of driving (EOD). When piles are driven into soft clay, a certain zone surrounding the clay becomes remolded or compressed, as shown in Figure 11.25a. This results in a reduction of undrained shear strength, cu (Figure 11.25b). With time, the loss of undrained shear strength is partially or fully regained. The time lapse may range from 30 to 60 days. For piles driven in dilative (dense to very dense) saturated fine sands, relaxation is possible. Negative pore water pressure, if developed during pile driving, will dissipate over time, resulting in a reduction in pile capacity with time after the driving operation is completed. At the same time, excess pore water pressure may be generated in contractive fine 0.5 D 1.5 D Pile Remolded zone Compressed zone D diameter 0.5 D 1.5 D (a) cu Remolded zone Sometime after driving Compressed zone Intact zone Immediately after driving Distance from pile (b) Figure 11.25 (a) Remolded or compacted zone around a pile driven into soft clay; (b) Nature of variation of undrained shear strength (cu ) with time around a pile driven into soft clay Intact zone 11.14 Pile Load Tests 587 sands during pile driving. The excess pore water pressure will dissipate over time, which will result in greater pile capacity. Several empirical relationships have been developed to predict changes in pile capacity with time. Skov and Denver (1988) Skov and Denver proposed the equation t Qt 5 QOED cA log a b 1 1d to where (11.68) Qt 5 pile capacity t days after the end of driving QOED 5 pile capacity at the end of driving t 5 time, in days For sand, A 5 0.2 and to 5 0.5 days; for clay, A 5 0.6 and to 5 1.0 days. Guang-Yu (1988) According to Guang-Yu, Q14 5 (0.375St 1 1)QOED where (applicable to clay soil) (11.69) Q14 5 pile capacity 14 days after pile driving St 5 sensitivity of clay Svinkin (1996) Svinkin suggests the relationship Qt 5 1.4QOEDt0.1 (upper limit for sand) Qt 5 1.025QOEDt0.1 (lower limit for sand) (11.70) (11.71) where t 5 time after driving, in days Example 11.9 Figure 11.26 shows the load test results of a 20-m long concrete pile (406 mm 3 406 mm) embedded in sand. Using Davisson’s method, determine the ultimate load Qu. Given: Ep 5 30 3 106 kN>m2. Solution From Eq. (11.67), su 5 0.012Dr 1 0.1a QuL D b1 Dr A pEp 588 Chapter 11: Pile Foundations Dr 5 300 mm, D 5 406 mm, L 5 20 m 5 20,000 mm, A p 5 406 mm 3 406 mm 5 164,836 mm2, and Ep 5 30 3 106 kN>m2. Hence, su 5 (0.012) (300) 1 (0.1) a (Qu ) (20,000) 406 b1 300 (30) (164,836) 5 3.6 1 0.135 1 0.004Qu 5 3.735 1 0.004Qu The line su (mm) 5 3.735 1 0.004Qu is drawn in Figure 11.26. The intersection of this line with the load-settlement curve gives the failure load Qu 5 1640 kN. 800 Qu = 1460 kN 1600 2400 Q (kN) 3.735 mm 5 10 15 20 Settlement, s (mm) Figure 11.26 11.15 ■ Elastic Settlement of Piles The total settlement of a pile under a vertical working load Qw is given by se 5 se(1) 1 se(2) 1 se(3) (11.72) 11.15 Elastic Settlement of Piles 589 where se(1) 5 elastic settlement of pile se(2) 5 settlement of pile caused by the load at the pile tip se(3) 5 settlement of pile caused by the load transmitted along the pile shaft If the pile material is assumed to be elastic, the deformation of the pile shaft can be evaluated, in accordance with the fundamental principles of mechanics of materials, as se(1) 5 (Qwp 1 jQws )L A pEp (11.73) where Qwp 5 load carried at the pile point under working load condition Qws 5 load carried by frictional (skin) resistance under working load condition A p 5 area of cross section of pile L 5 length of pile Ep 5 modulus of elasticity of the pile material The magnitude of j varies between 0.5 and 0.67 and will depend on the nature of the distribution of the unit friction (skin) resistance f along the pile shaft. The settlement of a pile caused by the load carried at the pile point may be expressed in the form: se(2) 5 qwpD Es (1 2 m2s )Iwp (11.74) where D 5 width or diameter of pile qwp 5 point load per unit area at the pile point 5 Qwp>A p Es 5 modulus of elasticity of soil at or below the pile point ms 5 Poisson’s ratio of soil Iwp 5 influence factor < 0.85 Vesic (1977) also proposed a semi-empirical method for obtaining the magnitude of the settlement of se(2) . His equation is se(2) 5 QwpCp Dqp where qp 5 ultimate point resistance of the pile Cp 5 an empirical coefficient Representative values of Cp for various soils are given in Table 11.13. (11.75) 590 Chapter 11: Pile Foundations Table 11.13 Typical Values of Cp [from Eq. (11.75)] Type of soil Driven pile Bored pile Sand (dense to loose) Clay (stiff to soft) Silt (dense to loose) 0.02–0.04 0.02–0.03 0.03–0.05 0.09–0.18 0.03–0.06 0.09–0.12 From “Design of Pile Foundations,” by A. S. Vesic. SYNTHESIS OF HIGHWAY PRACTICE by AMERICAN ASSOCIATION OF STATE HIGHWAY AND TRANSPORT. Copyright 1969 by TRANSPORTATION RESEARCH BOARD. Reproduced with permission of TRANSPORTATION RESEARCH BOARD in the format Textbook via Copyright Clearance Center. The settlement of a pile caused by the load carried by the pile shaft is given by a relation similar to Eq. (11.74), namely, se(3) 5 ¢ Qws D ≤ (1 2 m2s )Iws pL Es (11.76) where p 5 perimeter of the pile L 5 embedded length of pile Iws 5 influence factor Note that the term Qws>pL in Eq. (11.76) is the average value of f along the pile shaft. The influence factor, Iws, has a simple empirical relation (Vesic, 1977): L (11.77) ÅD Vesic (1977) also proposed a simple empirical relation similar to Eq. (11.75) for obtaining se(3) : Iws 5 2 1 0.35 se(3) 5 QwsCs Lqp In this equation, Cs 5 an empirical constant 5 (0.93 1 0.16"L>D)Cp (11.78) (11.79) The values of Cp for use in Eq. (11.75) may be estimated from Table 11.13. Example 11.10 The allowable working load on a prestressed concrete pile 21-m long that has been driven into sand is 502 kN. The pile is octagonal in shape with D 5 356 mm (see Table 11.3a). Skin resistance carries 350 kN of the allowable load, and point bearing carries the rest. Use Ep 5 21 3 106 kN>m2, Es 5 25 3 103 kN>m2, ms 0.35, and j 5 0.62. Determine the settlement of the pile. 11.16 Laterally Loaded Piles 591 Solution From Eq. (11.73), Se(1) 5 (Qwp 1 jQws )L A pEp From Table 11.3a for D 5 356 mm, the area of pile cross section. Ap 5 1045 cm2, Also, perimeter p 5 1.168 m. Given: Qws 5 350 kN, so Qwp 5 502 2 350 5 152 kN se(1) 5 3152 1 0.62(350)4 (21) (0.1045 m2 ) (21 3 106 ) 5 0.00353 m 5 3.35 mm From Eq. (11.74), qwpD 152 0.356 (1 2 m2s )Iwp 5 ¢ ≤¢ ≤ (1 2 0.352 ) (0.85) Es 0.1045 25 3 103 5 0.0155 m 5 15.5 mm se(2) 5 Again, from Eq. (11.76), se(3) 5 ¢ Qws D ≤ ¢ ≤ (1 2 m2s )Iws pL Es L 21 5 2 1 0.35 5 4.69 ÅD Å 0.356 350 0.356 5B R¢ ≤ (1 2 0.352 ) (4.69) (1.168) (21) 25 3 103 Iws 5 2 1 0.35 se(3) 5 0.00084 m 5 0.84 mm Hence, total settlement is se 5 se(1) 1 se(2) 1 se(3) 5 3.35 1 15.5 1 0.84 5 19.69 mm 11.16 ■ Laterally Loaded Piles A vertical pile resists a lateral load by mobilizing passive pressure in the soil surrounding it. (See Figure 11.1c.) The degree of distribution of the soil’s reaction depends on (a) the stiffness of the pile, (b) the stiffness of the soil, and (c) the fixity of the ends of the pile. In general, laterally loaded piles can be divided into two major categories: (1) short or rigid piles and (2) long or elastic piles. Figures 11.27a and 11.27b show the nature of the variation of the pile deflection and the distribution of the moment and shear force along the pile length when the pile is subjected to lateral loading. We next summarize the current solutions for laterally loaded piles. Elastic Solution A general method for determining moments and displacements of a vertical pile embedded in a granular soil and subjected to lateral load and moment at the ground surface was given by Matlock and Reese (1960). Consider a pile of length L subjected to a lateral force Qg and 592 Chapter 11: Pile Foundations Deflection Qg Shear Moment Mg (a) Loading Deflection Moment Qg Mg Shear z (b) Figure 11.27 Nature of variation of pile deflection, moment, and shear force for (a) a rigid pile and (b) and elastic pile Mg Qg x p L z z (b) (a) x x x x x x p V M z z z z z (c) Figure 11.28 (a) Laterally loaded pile; (b) soil resistance on pile caused by lateral load; (c) sign conventions for displacement, slope, moment, shear, and soil reaction 11.16 Laterally Loaded Piles 593 a moment Mg at the ground surface (z 5 0), as shown in Figure 11.28a. Figure 11.28b shows the general deflected shape of the pile and the soil resistance caused by the applied load and the moment. According to a simpler Winkler’s model, an elastic medium (soil in this case) can be replaced by a series of infinitely close independent elastic springs. Based on this assumption, pr(kN>m) k5 (11.80) x(m) where k 5 modulus of subgrade reaction pr 5 pressure on soil x 5 deflection The subgrade modulus for granular soils at a depth z is defined as kz 5 nhz (11.81) where nh 5 constant of modulus of horizontal subgrade reaction. Referring to Figure 11.28b and using the theory of beams on an elastic foundation, we can write d4x EpIp 4 5 pr (11.82) dz where Ep 5 modulus of elasticity in the pile material Ip 5 moment of inertia of the pile section Based on Winkler’s model pr 5 2kx (11.83) The sign in Eq. (11.83) is negative because the soil reaction is in the direction opposite that of the pile deflection. Combining Eqs. (11.82) and (11.83) gives EpIp d4x 1 kx 5 0 dz4 (11.84) The solution of Eq. (11.84) results in the following expressions: Pile Deflection at Any Depth [xz(z)] xz (z) 5 A x QgT3 EpIp 1 Bx MgT2 EpIp (11.85) Slope of Pile at Any Depth [uz(z)] uz (z) 5 A u QgT2 EpIp 1 Bu MgT EpIp (11.86) 594 Chapter 11: Pile Foundations Moment of Pile at Any Depth [Mz(z)] Mz (z) 5 A mQgT 1 BmMg (11.87) Shear Force on Pile at Any Depth [Vz(z)] Vz (z) 5 A vQg 1 Bv Mg T (11.88) Soil Reaction at Any Depth [ p z9 (z)] pzr (z) 5 A pr Qg T 1 Bpr Mg T2 (11.89) where A x , Bx , A u , Bu , A m , Bm , A v , Bv , A pr , and Bpr are coefficients T 5 characteristic length of the soil–pile system 5 EpIp Å nh 5 (11.90) nh has been defined in Eq. (11.81) When L $ 5T, the pile is considered to be a long pile. For L # 2T, the pile is considered to be a rigid pile. Table 11.14 gives the values of the coefficients for long piles (L>T $ 5) in Eqs. (11.85) through (11.89). Note that, in the first column of the table, Z5 z T (11.91) is the nondimensional depth. The positive sign conventions for xz (z), uz (z), Mz (z), Vz (z), and pzr (z) assumed in the derivations in Table 11.14 are shown in Figure 11.28c. Figure 11.29 shows the variation of A x , Bx , A m , and Bm for various values of L>T 5 Zmax . It indicates that, when L>T is greater than about 5, the coefficients do not change, which is true of long piles only. Calculating the characteristic length T for the pile requires assuming a proper value of nh . Table 11.15 gives some representative values. Elastic solutions similar to those given in Eqs. 11.85 through 11.89 for piles embedded in cohesive soil were developed by Davisson and Gill (1963). Their equations are xz (z) 5 A xr QgR3 EpIp 1 Bxr MgR2 EpIp (11.92) 11.16 Laterally Loaded Piles 595 Table 11.14 Coefficients for Long Piles, kz 5 nhz Z 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 3.0 4.0 5.0 Ax 2.435 2.273 2.112 1.952 1.796 1.644 1.496 1.353 1.216 1.086 0.962 0.738 0.544 0.381 0.247 0.142 20.075 20.050 20.009 Au 21.623 21.618 21.603 21.578 21.545 21.503 21.454 21.397 21.335 21.268 21.197 21.047 20.893 20.741 20.596 20.464 20.040 0.052 0.025 Am 0.000 0.100 0.198 0.291 0.379 0.459 0.532 0.595 0.649 0.693 0.727 0.767 0.772 0.746 0.696 0.628 0.225 0.000 20.033 Av 1.000 0.989 0.956 0.906 0.840 0.764 0.677 0.585 0.489 0.392 0.295 0.109 20.056 20.193 20.298 20.371 20.349 20.106 0.015 Ap9 Bx Bu 0.000 20.227 20.422 20.586 20.718 20.822 20.897 20.947 20.973 20.977 20.962 20.885 20.761 20.609 20.445 20.283 0.226 0.201 0.046 1.623 1.453 1.293 1.143 1.003 0.873 0.752 0.642 0.540 0.448 0.364 0.223 0.112 0.029 20.030 20.070 20.089 20.028 0.000 21.750 21.650 21.550 21.450 21.351 21.253 21.156 21.061 20.968 20.878 20.792 20.629 20.482 20.354 20.245 20.155 0.057 0.049 20.011 Bm 1.000 1.000 0.999 0.994 0.987 0.976 0.960 0.939 0.914 0.885 0.852 0.775 0.688 0.594 0.498 0.404 0.059 20.042 20.026 Bv 0.000 20.007 20.028 20.058 20.095 20.137 20.181 20.226 20.270 20.312 20.350 20.414 20.456 20.477 20.476 20.456 20.213 0.017 0.029 Bp9 0.000 20.145 20.259 20.343 20.401 20.436 20.451 20.449 20.432 20.403 20.364 20.268 20.157 20.047 0.054 0.140 0.268 0.112 20.002 From Drilled Pier Foundations, by R. J. Woodward, W. S. Gardner, and D. M. Greer. Copyright 1972 McGraw-Hill. Used with permission of the McGraw-Hill Book Company. Table 11.15 Representative Values of nh Soil Dry or moist sand Loose Medium Dense Submerged sand Loose Medium Dense nh kN>m3 1800–2200 5500–7000 15,000–18,000 1000–1400 3500–4500 9000–12,000 and Mz (z) 5 A m r QgR 1 Bm r Mg (11.93) where A xr , Bx , A m r , and Bm r are coefficients. and R5 EpIp É k 4 (11.94) 596 Chapter 11: Pile Foundations –1 0 Ax 2 1 3 4 5 0 4 3 Zmax 2 Z z/T 1 2 3 4 4 10 (a) 5 Bx –1 0 1 2 3 4 0 4 3 Zmax 2 1 Z z/T 2 3 Figure 11.29 Variation of A x , Bx , A m , and Bm with Z (From Matlock, H. and Reese, L. C. (1960). “Generalized Solution for Laterally Loaded Piles,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 86, No. SM5, Part I, pp. 63–91. With permission from ASCE.) 4, 10 4 5 (b) The values of the coefficients Ar and Br are given in Figure 11.30 Note that Z5 z R (11.95) and Zmax 5 L R (11.96) 11.16 Laterally Loaded Piles 0 0.2 Am 0.6 0.4 0.8 1.0 597 1.2 0 Z z/T 1 Zmax 2 2 3 4 3 10 4 5 (c) Bm 0 0.2 0.4 0.6 0.8 1.0 0 1 Zmax 2 Z z/T 2 3 4 3 10 4 5 (d) Figure 11.29 (continued) The use of Eqs. (11.92) and (11.93) requires knowing the magnitude of the characteristic length, R. This can be calculated from Eq. (11.94), provided that the coefficient of the subgrade reaction is known. For sands, the coefficient of the subgrade reaction was given by Eq. (11.81), which showed a linear variation with depth. However, in cohesive 598 Chapter 11: Pile Foundations Ax, Am –1 0 1 2 0 2 3 Zmax 5 1 2 Zmax 2 Z 3 4 5 3 4 Ax Am 5 –2 (a) Bx, Bm 0 –1 0 1 2 4, 5 Zmax 2 1 Zmax 2 3 Z 2 4 3 3 5 r , and Figure 11.30 Variation of A x , Bxr , A m Bm r with Z (From Davisson, M. T. and Gill, H. L. (1963). “Laterally Loaded Piles in a Layered Soil System,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 89, No. SM3, pp. 63–94. With permission from ASCE.) 4 Bx Bm 5 (b) 11.16 Laterally Loaded Piles 599 soils, the subgrade reaction may be assumed to be approximately constant with depth. Vesic (1961) proposed the following equation to estimate the value of k: EsD4 Es k 5 0.65 12 Å EpIp 1 2 m2s (11.97) Here, Es 5 modulus of elasticity of soil D 5 pile width (or diameter) ms 5 Poisson’s ratio for the soil For all practical puroses, Eq. (11.97) can be written as k< Es (11.98) 1 2 m2s Ultimate Load Analysis: Broms’s Method For laterally loaded piles, Broms (1965) developed a simplified solution based on the assumptions of (a) shear failure in soil, which is the case for short piles, and (b) bending of the pile, governed by the plastic yield resistance of the pile section, which is applicable to long piles. Broms’s solution for calculating the ultimate load resistance, Qu(g) , for short piles is given in Figure 11.31a. A similar solution for piles embedded in cohesive soil is shown in Figure 11.31b. In Figure 11.31a, note that Qu(g) L e D L 0.4 0.2 160 120 Free-headed pile 8 0. 80 5 1.0 1. 0 . 2 3.0 Restrained pile 40 4 2 Restrained pile cuD2 Ultimate lateral resistance, Qu(g) e L 0 50 0.6 Qu(g) KpD3 200 Ultimate lateral resistance, e D 0 1 60 Free-headed pile Restrained pile 40 Free-headed pile 8 Qu(g) 30 16 20 10 0 0 0 4 8 12 L Length, D (a) 16 20 0 4 8 12 16 20 L Embedment length, D (b) Figure 11.31 Broms’s solution for ultimate lateral resistance of short piles (a) in sand and (b) in clay 600 Chapter 11: Pile Foundations Kp 5 Rankine passive earth pressure coefficient 5 tan2 ¢ 45 1 fr ≤ 2 (11.99) Similarly, in Figure 11.31b, cu 5 undrained cohesion < 0.75qu 0.75qu 5 5 0.375qu FS 2 (11.100) where FS 5 factor of safety(52) qu 5 unconfined compression strength Figure 11.32 shows Broms’s analysis of long piles. In the figure, the yield moment for the pile is My 5 SFY (11.101) where S 5 section modulus of the pile section FY 5 yield stress of the pile material In solving a given problem, both cases (i.e., Figure 11.31 and Figure 11.32) should be checked. The deflection of the pile head, xz (z 5 0), under working load conditions can be estimated from Figure 11.33. In Figure 11.33a, the term h can be expressed as h5 nh 5 Å EpIp (11.102) The range of nh for granular soil is given in Table 11.15. Similarly, in Figure 11.33b, which is for clay, the term K is the horizontal soil modulus and can be defined as K5 pressure (kN>m2 ) displacement (m) (11.103) KD Å 4EpIp (11.104) Also, the term b can be defined as b5 4 Note that, in Figure 11.33, Qg is the working load. The following is a general range of values of K for clay soils. Unconfined compression strength, qu kN>m2 200 200–800 . 800 K kN>m3 10,000–20,000 20,000–40,000 . 40,000 11.16 Laterally Loaded Piles Free-headed pile 100 Restrained pile 10 e 0 D 1 2 4 8 16 32 Qu(g) Ultimate lateral resistance, KpD3 1000 1 0.10 1.0 10.0 Yield moment, (a) 100.0 My 1000.0 10,000.0 D4Kp 100 60 Restrained pile cuD2 20 e 0 D 10 6 Free-headed pile 8 4 1 4 2 Ultimate lateral resistance, Qu(g) 40 16 2 1 3 4 6 10 20 40 60 Yield moment, (b) 100 200 400 600 My cuD3 Figure 11.32 Broms’s solution for ultimate lateral resistance of long piles (a) in sand (b) in clay 601 602 Chapter 11: Pile Foundations Dimensionless lateral deflection, xz(z 0)(Ep Ip )3/5(nh )2/5 Qg L 10 8 6 Restrained pile Free-headed pile e 2.0 L 4 1.5 2 1.0 0.8 0.6 0.4 0.2 0.0 0 0 2 4 6 8 10 Dimensionless length, L (a) 10 e 0.4 L Dimensionless lateral deflection, 0.2 0.1 xz(z 0)KDL Qg 8 6 5 0.0 0.0 4 Free-headed pile Restrained pile 2 0 0 1 2 3 4 Dimensionless length, L (b) Figure 11.33 Broms’s solution for estimating deflection of pile head (a) in sand and (b) in clay 5 11.16 Laterally Loaded Piles 603 Example 11.11 Consider a steel H-pile (HP 250 3 85) 25 m long, embedded fully in a granular soil. Assume that nh 5 12,000 kN>m3 . The allowable displacement at the top of the pile is 8 mm. Determine the allowable lateral load, Qg . Let Mg 5 0. Use the elastic solution. Solution From Table 11.1a, for an HP 250 3 85 pile, Ip 5 123 3 1026 m4 (about the strong axis) and let Ep 5 207 3 106 kN>m2 From Eq. (11.90), T5 EpIp Å nh 5 5 (207 3 106 ) (123 3 1026 ) 5 1.16 m Å 12,000 5 Here, L>T 5 25>1.16 5 21.55 . 5, so the pile is a long one. Because Mg 5 0, Eq. (11.85) takes the form xz (z) 5 A x QgT3 EpIp and it follows that Qg 5 xz (z)EpIp A xT3 At z 5 0, xz 5 8 mm 5 0.008 m and A x 5 2.435 (see Table 11.14), so Qg 5 (0.008) (207 3 106 ) (123 3 1026 ) (2.435) (1.163 ) 5 53.59 kN This magnitude of Qg is based on the limiting displacement condition only. However, the magnitude of Qg based on the moment capacity of the pile also needs to be determined. For Mg 5 0, Eq. (11.87) becomes Mz (z) 5 A mQgT According to Table 11.14, the maximum value of A m at any depth is 0.772. The maximum allowable moment that the pile can carry is Mz(max) 5 FY Ip d1 2 604 Chapter 11: Pile Foundations Let FY 5 248,000 kN>m2 . From Table 11.1a, Ip 5 123 3 1026 m4 and d1 5 0.254 m, so Ip d1 ¢ ≤ 2 5 123 3 1026 0.254 ¢ ≤ 2 5 968.5 3 1026 m3 Now, Qg 5 Mz(max) A mT 5 (968.5 3 1026 ) (248,000) 5 268.2 kN (0.772) (1.16) Because Qg 5 268.2 kN . 53.59 kN, Qg 5 53.59 kN. the deflection criteria apply. Hence, ■ Example 11.12 Solve Example 11.11 by Broms’s method. Assume that the pile is flexible and is free headed. Let the yield stress of the pile material, Fy 5 248 MN>m2 ; the unit weight of soil, g 5 18 kN>m3 ; and the soil friction angle fr 5 35°. Solution We check for bending failure. From Eq. (11.101), My 5 SFy From Table 11.1a, S5 Ip d1 2 5 123 3 1026 0.254 2 Also, My 5 123 3 1026 (248 3 103 ) 5 240.2 kN-m £ 0.254 § 2 and My 4 D gKp 5 My fr D g tan ¢ 45 1 ≤ 2 4 2 5 240.2 35 (0.254) (18) tan ¢ 45 1 ≤ 2 4 5 868.8 2 From Figure 11.32a, for My>D4gKp 5 868.8, the magnitude of Qu(g)>KpD3g (for a free-headed pile with e>D 5 0) is about 140, so Qu(g) 5 140KpD3g 5 140 tan2 ¢45 1 35 ≤ (0.254) 3 (18) 5 152.4 kN 2 11.16 Laterally Loaded Piles 605 Next, we check for pile head deflection. From Eq. (11.102), h5 nh 12,000 5 5 5 0.86 m21 Å EpIp Å (207 3 106 ) (123 3 1026 ) 5 so hL 5 (0.86) (25) 5 21.5 From Figure 11.33a, for hL 5 21.5, e>L 5 0 (free-headed pile): thus, xo (EpIp ) 3>5 (nh ) 2>5 QgL < 0.15 (by interpolation) and Qg 5 5 xo (EpIp ) 3>5 (nh ) 2>5 0.15L (0.008) 3(207 3 106 ) (123 3 1026 )4 3>5 (12,000) 2>5 (0.15) (25) Hence, Qg 5 40.2 kN (*152.4 kN). 5 40.2 kN ■ Example 11.13 Assume that the 25-m long pile described in Example 11.11 is a restrained pile and is embedded in clay soil. Given: cu 5 100 kN>m2 and K 5 5,000 kN>m3. The allowable lateral displacement at the top of the pile is 10 mm. Determine the allowable lateral load Qg. Given Mymg 5 0. Use Broms’s method. Solution From Example 11.12, My 5 240.2 kN-m. So My cuD3 5 240.2 5 146.6 (100) (0.254) 3 For the unrestrained pile, from Figure 11.32b, Qu(g) cuD2 < 65 or Qu(g) 5 (65) (100) (0.254) 2 5 419.3 kN 606 Chapter 11: Pile Foundations Check Pile-Head Deflection From Eq. (11.104), b5 KD (5000) (0.254) 4 5 4 5 0.334 Å 4EpIp Å (4) (207 3 106 ) (123 3 1026 ) bL 5 (0.334) (25) 5 8.35 From Figure 11.33b for bL 5 8.35, by extrapolation the magnitude of xz (z 5 0)KDL Qg Qg 5 xz (z 5 0)KDL 8 5 a <8 10 b (5000) (0.254) (25) 1000 5 39.7 kN 8 Hence, Qg 5 39.7 kN(* 419.3 kN) . 11.17 ■ Pile-Driving Formulas To develop the desired load-carrying capacity, a point bearing pile must penetrate the dense soil layer sufficiently or have sufficient contact with a layer of rock. This requirement cannot always be satisfied by driving a pile to a predetermined depth, because soil profiles vary. For that reason, several equations have been developed to calculate the ultimate capacity of a pile during driving. These dynamic equations are widely used in the field to determine whether a pile has reached a satisfactory bearing value at the predetermined depth. One of the earliest such equations—commonly referred to as the Engineering News (EN) Record formula—is derived from the work—energy theory. That is, Energy imparted by the hammer per blow 5 (pile resistance)(penetration per hammer blow) According to the EN formula, the pile resistance is the ultimate load Qu , expressed as Qu 5 where WR 5 weight of the ram h 5 height of fall of the ram S 5 penetration of pile per hammer blow C 5 a constant WRh S1C (11.105) 11.17 Pile-Driving Formulas 607 The pile penetration, S, is usually based on the average value obtained from the last few driving blows. In the equation’s original form, the following values of C were recommended: For drop hammers, C 5 25.4 mm if S and h are in mm For steam hammers, C 5 2.54 mm if S and h are in mm Also, a factor of safety FS 5 6 was recommended for estimating the allowable pile capacity. Note that, for single- and double-acting hammers, the term WRh can be replaced by EHE , where E is the efficiency of the hammer and HE is the rated energy of the hammer. Thus, Qu 5 EHE S1C (11.106) The EN formula has been revised several times over the years, and other pile-driving formulas also have been suggested. Three of the other relationships generally used are tabulated in Table 11.16. The maximum stress developed on a pile during the driving operation can be estimated from the pile-driving formulas presented in Table 11.16. To illustrate, we use the modified EN formula: Qu 5 2 EWRh WR 1 n Wp S 1 C WR 1 Wp In this equation, S is the average penetration per hammer blow, which can also be expressed as S5 25.4 N (11.107) where S is in mm N 5 number of hammer blows per 25.4 mm of penetration Thus, WR 1 n2Wp EWRh Qu 5 (25.4>N) 1 2.54 WR 1 Wp (11.108) Different values of N may be assumed for a given hammer and pile, and Qu may be calculated. The driving stress Qu>A p can then be calculated for each value of N. This 608 Chapter 11: Pile Foundations Table 11.16 Pile-Driving Formulas Name Modified EN formula Formula 2 EWRh WR 1 n Wp S 1 C WR 1 Wp where E 5 efficiency of hammer C 5 2.54 mm if the units of S and h are in mm Wp 5 weight of the pile n 5 coefficient of restitution between the ram and the pile cap Qu 5 Typical values for E Single- and double-acting hammers Diesel hammers Drop hammers 0.7–0.85 0.8–0.9 0.7–0.9 Typical values for n Cast-iron hammer and concrete piles (without cap) Wood cushion on steel piles Wooden piles Danish formula (Olson and Flaate, 1967) EHE Qu 5 S1 where Janbu’s formula (Janbu, 1953) Qu 5 where 0.4–0.5 0.3–0.4 0.25–0.3 EHEL É 2A pEp E 5 efficiency of hammer HE 5 rated hammer energy Ep 5 modulus of elasticity of the pile material L 5 length of the pile A p 5 cross-sectional area of the pile EHE Kur S Kur 5 Cd ¢1 1 Å 11 Cd 5 0.75 1 0.14 ¢ lr 5 ¢ EHEL A pEpS2 ≤ lr ≤ Cd Wp WR ≤ 11.17 Pile-Driving Formulas 609 procedure can be demonstrated with a set of numerical values. Suppose that a prestressed concrete pile 24.4 m in length has to be driven by a hammer. The pile sides measure 254 mm. From Table 11.3a, for this pile, A p 5 645 3 1024 m2 The weight of the pile is A pLgc 5 (645 3 1024 ) (24.4 m) (23.58 kN>m3 ) 5 37.1 kN If the weight of the cap is 2.98 kN, then Wp 5 37.1 1 2.98 5 40.08 kN For the hammer, let Rated energy 5 26.03 kN-m 5 HE 5 WRh Weight of ram 5 22.24 kN Assume that the hammer efficiency is 0.85 and that n 5 0.35. Substituting these values into Eq. (11.108) yields Qu 5 (0.85) (26.03 3 1000) 22.24 1 (0.35) 2 (40.08) 9639.08 c d 5 kip £ § 25.4 22.24 1 40.08 25.4 1 2.54 1 2.54 N N Now the following table can be prepared: N Qu (kN) 0 2 4 6 8 10 12 20 0 632 1084 1423 1687 1898 2070 2530 Ap (m2) 645 3 645 3 645 3 645 3 645 3 645 3 645 3 645 3 10 24 10 24 10 24 10 24 10 24 10 24 10 24 10 24 Qu/Ap (MN/m2) 0 9.79 16.8 22.06 26.16 29.43 32.12 39.22 Both the number of hammer blows per inch and the stress can be plotted in a graph, as shown in Figure 11.34. If such a curve is prepared, the number of blows per inch of pile penetration corresponding to the allowable pile-driving stress can easily be determined. Actual driving stresses in wooden piles are limited to about 0.7fu . Similarly, for concrete and steel piles, driving stresses are limited to about 0.6fcr and 0.85fy , respectively. In most cases, wooden piles are driven with a hammer energy of less than 60 kN-m. Driving resistances are limited mostly to 4 to 5 blows per inch of pile penetration. For concrete and steel piles, the usual values of N are 6 to 8 and 12 to 14, respectively. 610 Chapter 11: Pile Foundations 40 Qu /Ap (MN/m2) 30 20 10 0 0 4 8 12 16 Number of blows /25.4 mm (N) 20 Figure 11.34 Plot of stress versus blows>25.4 mm. Example 11.14 A precast concrete pile 0.305 m 3 0.305 m in cross section is driven by a hammer. Given Maximum rated hammer energy 5 40.67 kN-m Hammer efficiency 5 0.8 Weight of ram 5 33.36 kN Pile length 5 24.39 m Coefficient of restitution 5 0.4 Weight of pile cap 5 2.45 kN Ep 5 20.7 3 106 kN>m2 Number of blows for last 25.4 mm of penetration 5 8 Estimate the allowable pile capacity by the a. Modified EN formula (use FS 5 6) b. Danish formula (use FS 5 4) Solution Part a Qu 5 2 EWRh WR 1 n Wp S 1 C WR 1 Wp Weight of pile 1 cap 5 (0.305 3 0.305 3 24.39) (23.58 kN>m3 ) 1 2.45 5 55.95 kN 11.18 Pile Capacity For Vibration-Driven Piles 611 Given: WRh 5 40.67 kN-m Qu 5 Qall 5 (0.8) (40.67 3 1000) 25.4 8 3 1 2.54 33.36 1 (0.4) 2 (55.95) 5 2697 kN 33.36 1 55.95 Qu 2697 5 < 449.5 kN FS 6 Part b Qu 5 EHE EHEL S1 Ä 2A pEp Use Ep 5 20.7 3 106 kN>m2 EHEL (0.8) (40.67) (24.39) 5 5 0.01435 m 5 14.35 mm Å 2A pEp Å 2(0.305 3 0.305) (20.7 3 106 kN>m2 ) Qu 5 Qall 5 11.18 (0.8) (40.67) 25.4 8 3 1000 1 0.01435 < 1857 kN 1857 < 464 kN 4 ■ Pile Capacity For Vibration-Driven Piles The principles of vibratory pile drivers (Figure 11.7e) were discussed briefly in Section 11.4. As mentioned there, the driver essentially consists of two counterrotating weights. The amplitude of the centrifugal driving force generated by a vibratory hammer can be given as Fc 5 mev2 (11.109) where m 5 total eccentric rotating mass e 5 distance between the center of each rotating mass and the center of rotation v 5 operating circular frequency Vibratory hammers typically include an isolated bias weight that can range from 4 to 40 kN. The bias weight is isolated from oscillation by springs, so it acts as a net downward load helping the driving efficiency by increasing the penetration rate of the pile. The use of vibratory pile drivers began in the early 1930s. Installing piles with vibratory drivers produces less noise and damage to the pile, compared with impact driving. However, because of a limited understanding of the relationships between the load, the rate of penetration, and the bearing capacity of piles, this method has not gained popularity in the United States. 612 Chapter 11: Pile Foundations Vibratory pile drivers are patented. Some examples are the Bodine Resonant Driver (BRD), the Vibro Driver of the McKiernan-Terry Corporation, and the Vibro Driver of the L. B. Foster Company. Davisson (1970) provided a relationship for estimating the ultimate pile capacity in granular soil: In SI units, Qu (kN) 5 0.746(Hp ) 1 98(vp m>s) (vp m>s) 1 (SL m>cycle) (fHz) (11.110) where Hp 5 horsepower delivered to the pile vp 5 final rate of pile penetration SL 5 loss factor f 5 frequency, in Hz The loss factor SL for various types of granular soils is as follows (Bowles, 1996): Closed-End Pipe Piles • • • Loose sand: 0.244 3 1023 m>cycle Medium dense sand: 0.762 3 1023 m>cycle Dense sand: 2.438 3 1023 m>cycle H-Piles • • • Loose sand: 20.213 3 1023 m>cycle Medium dense sand: 0.762 3 1023 m>cycle Dense sand: 2.134 3 1023 m>cycle In 2000, Feng and Deschamps provided the following relationship for the ultimate capacity of vibrodriven piles in granular soil: Qu 5 3.6(Fc 1 11WB ) LE vp L 1 1 1.8 3 1010 "OCR c Here, Fc 5 centrifugal force WB 5 bias weight vp 5 final rate of pile penetration c 5 speed of light 31.8 3 1010 m>min4 OCR 5 overconsolidation ratio LE 5 embedded length of pile L 5 pile length (11.111) 11.19 Negative Skin Friction 613 Example 11.15 Consider a 20-m-long steel pile driven by a Bodine Resonant Driver (Section HP 310 3 125) in a medium dense sand. If Hp 5 350 horsepower, vp 5 0.0016 m>s, and f 5 115 Hz, calculate the ultimate pile capacity, Qu . Solution From Eq. (11.110), Qu 5 0.746Hp 1 98vp vp 1 SL f For an HP pile in medium dense sand, SL < 0.762 3 1023 m>cycle. So Qu 5 11.19 (0.746) (350) 1 (98) (0.0016) 0.0016 1 (0.762 3 1023 ) (115) 5 2928 kN ■ Negative Skin Friction Negative skin friction is a downward drag force exerted on a pile by the soil surrounding it. Such a force can exist under the following conditions, among others: 1. If a fill of clay soil is placed over a granular soil layer into which a pile is driven, the fill will gradually consolidate. The consolidation process will exert a downward drag force on the pile (see Figure 11.35a) during the period of consolidation. 2. If a fill of granular soil is placed over a layer of soft clay, as shown in Figure 11.35b, it will induce the process of consolidation in the clay layer and thus exert a downward drag on the pile. 3. Lowering of the water table will increase the vertical effective stress on the soil at any depth, which will induce consolidation settlement in clay. If a pile is located in the clay layer, it will be subjected to a downward drag force. Sand H f fill Clay H f fill z L L Sand L1 Neutral plane z Clay (a) Figure 11.35 Negative skin friction (b) 614 Chapter 11: Pile Foundations In some cases, the downward drag force may be excessive and cause foundation failure. This section outlines two tentative methods for the calculation of negative skin friction. Clay Fill over Granular Soil (Figure 11.35a) Similar to the b method presented in Section 11.12, the negative (downward) skin stress on the pile is fn 5 Krsor tan dr (11.112) where Kr 5 earth pressure coefficient 5 Ko 5 1 2 sin fr sor 5 vertical effective stress at any depth z 5 gfr z gfr 5 effective unit weight of fill dr 5 soil–pile friction angle < 0.5–0.7fr Hence, the total downward drag force on a pile is Hf Qn 5 3 (pKrgfr tan dr)z dz 5 pKrgfr H 2f tan dr 2 0 (11.113) where Hf 5 height of the fill. If the fill is above the water table, the effective unit weight, gfr , should be replaced by the moist unit weight. Granular Soil Fill over Clay (Figure 11.35b) In this case, the evidence indicates that the negative skin stress on the pile may exist from z 5 0 to z 5 L1 , which is referred to as the neutral depth. (See Vesic, 1977, pp. 25 –26.) The neutral depth may be given as (Bowles, 1982) L1 5 (L 2 Hf ) L1 B L 2 Hf 2 1 gfr Hf gr R 2 2gfr Hf gr (11.114) where gfr and gr 5 effective unit weights of the fill and the underlying clay layer, respectively. For end-bearing piles, the neutral depth may be assumed to be located at the pile tip (i.e., L1 5 L 2 Hf). Once the value of L1 is determined, the downward drag force is obtained in the following manner: The unit negative skin friction at any depth from z 5 0 to z 5 L1 is fn 5 Krsor tan dr where Kr 5 Ko 5 1 2 sin fr sor 5 gfr Hf 1 grz dr 5 0.5–0.7fr (11.115) 11.19 Negative Skin Friction L1 615 L1 Qn 5 3 pfn dz 5 3 pKr(gfr Hf 1 grz)tan dr dz 0 0 5 (pKrgfr Hf tan dr)L1 1 L21pKrgr tan dr 2 (11.116) If the soil and the fill are above the water table, the effective unit weights should be replaced by moist unit weights. In some cases, the piles can be coated with bitumen in the downdrag zone to avoid this problem. A limited number of case studies of negative skin friction is available in the literature. Bjerrum et al. (1969) reported monitoring the downdrag force on a test pile at Sorenga in the harbor of Oslo, Norway (noted as pile G in the original paper). The study of Bjerrum et al. (1969) was also discussed by Wong and Teh (1995) in terms of the pile being driven to bedrock at 40 m. Figure 11.36a shows the soil profile and the pile. Wong and Teh estimated the following quantities: Fill: Moist unit weight, gf 5 16 kN>m3 Saturated unit weight, gsat(f) 5 18.5 kN>m3 So gfr 5 18.5 2 9.81 5 8.69 kN>m3 and Hf 5 13 m • f 16 kN/m3 2m 11 m 0 0 Fill Groundwater table Axial force in pile (kN) 1000 2000 3000 Fill sat ( f ) 18.5 kN/m3 40 m Pile D 500 mm Depth (m) 10 20 30 Clay 40 Rock (a) (b) Figure 11.36 Negative skin friction on a pile in the harbor of Oslo, Norway (Based on Bjerrum et al., (1969) and Wong and The (1995)) 616 Chapter 11: Pile Foundations Clay: Kr tan dr < 0.22 Saturated effective unit weight, gr 5 19 2 9.81 5 9.19 kN>m3 Pile: L 5 40 m Diameter, D 5 500 m • • Thus, the maximum downdrag force on the pile can be estimated from Eq. (11.116). Since in this case the pile is a point bearing pile, the magnitude of L1 5 27 m, and Qn 5 (p) (Kr tan dr) 3gf 3 2 1 (13 2 2)gfr 4 (L1 ) 1 L21pgr(Kr tan dr) 2 or Qn 5 (p 3 0.5)(0.22) 3(16 3 2) 1 (8.69 3 11)4 (27) 1 (27) 2 (p 3 0.5)(9.19)(0.22) 2 5 2348 kN The measured value of the maximum Qn was about 2500 kN (Figure 11.36b), which is in good agreement with the calculated value. Example 11.16 In Figure 11.35a, let Hf 5 2m. The pile is circular in cross section with a diameter of 0.305 m. For the fill that is above the water table, gf 5 16 kN>m3 and fr 5 32°. Determine the total drag force. Use dr 5 0.6fr. Solution From Eq. (11.113), Qn 5 pK rgfH 2f tan dr 2 with p 5 p(0.305) 5 0.958m K r 5 1 2 sin fr 5 1 2 sin 32 5 0.47 and dr 5 (0.6) (32) 5 19.2° Thus, Qn 5 (0.958) (0.47) (16) (2) 2 tan 19.2 5 5.02 kN 2 ■ Example 11.17 In Figure 11.35b, let Hf 5 2 m, pile diameter 5 0.305 m, gf 5 16.5 kN>m3, fclay r 5 34°, gsat(clay) 5 17.2 kN>m3, and L 5 20 m. The water table coincides with the r . top of the clay layer. Determine the downward drag force. Assume that dr 5 0.6fclay 11.20 Group Efficiency 617 Solution The depth of the neutral plane is given in Eq. (11.114) as L1 5 L 2 Hf L1 a L 2 Hf 2 1 gfHf gr b2 2gfHf gr Note that gfr in Eq. (11.114) has been replaced by gf because the fill is above the water table, so L1 5 (20 2 2) (20 2 2) (16.5) (2) (2) (16.5) (2) c 1 d 2 L1 2 17.2 2 9.81) (17.2 2 9.81) or L1 5 242.4 2 8.93; L1 5 11.75 m L1 Now, from Eq. (11.116), we have Qn 5 (pK rgfHf tan dr )L1 1 L21pK rgr tan dr 2 with p 5 p(0.305) 5 0.958m and K r 5 1 2 sin 34° 5 0.44 Hence, Qn 5 (0.958) (0.44) (16.5) (2) 3tan(0.6 3 34)4 (11.75) 1 (11.75) 2 (0.958) (0.44) (17.2 2 9.81) 3tan(0.6 3 34) 4 5 60.78 1 79.97 5 140.75 kN 2 ■ Group Piles 11.20 Group Efficiency In most cases, piles are used in groups, as shown in Figure 11.37, to transmit the structural load to the soil. A pile cap is constructed over group piles. The cap can be in contact with the ground, as in most cases (see Figure 11.37a), or well above the ground, as in the case of offshore platforms (see Figure 11.37b). Determining the load-bearing capacity of group piles is extremely complicated and has not yet been fully resolved. When the piles are placed close to each other, a 618 Chapter 11: Pile Foundations Pile cap Section Water table L d d d d L Plan (b) d Bg Lg d Number of piles in group n1 n2 (Note: Lg Bg) Lg (n1 1) d 2(D/2) Bg (n2 1) d 2(D/2) (a) (c) Figure 11.37 Group piles reasonable assumption is that the stresses transmitted by the piles to the soil will overlap (see Figure 11.37c), reducing the load-bearing capacity of the piles. Ideally, the piles in a group should be spaced so that the load-bearing capacity of the group is not less than the sum of the bearing capacity of the individual piles. In practice, the minimum centerto-center pile spacing, d, is 2.5D and, in ordinary situations, is actually about 3 to 3.5D. 11.20 Group Efficiency 619 The efficiency of the load-bearing capacity of a group pile may be defined as h5 Qg(u) S Qu (11.117) where h 5 group efficiency Qg(u) 5 ultimate load-bearing capacity of the group pile Qu 5 ultimate load-bearing capacity of each pile without the group effect Many structural engineers use a simplified analysis to obtain the group efficiency for friction piles, particularly in sand. This type of analysis can be explained with the aid of Figure 11.37a. Depending on their spacing within the group, the piles may act in one of two ways: (1) as a block, with dimensions Lg 3 Bg 3 L, or (2) as individual piles. If the piles act as a block, the frictional capacity is favpgL < Qg(u) . [Note: pg 5 perimeter of the cross section of block 5 2(n1 1 n2 2 2)d 1 4D, and fav 5 average unit frictional resistance.] Similarly, for each pile acting individually, Qu < pLfav . (Note: p 5 perimeter of the cross section of each pile.) Thus, h5 5 Qg(u) S Qu 5 fav 32(n1 1 n2 2 2)d 1 4D4L n1n2pLfav 2(n1 1 n2 2 2)d 1 4D pn1n2 (11.118) Hence, Qg(u) 5 B 2(n1 1 n2 2 2)d 1 4D RS Qu pn1n2 (11.119) From Eq. (11.119), if the center-to-center spacing d is large enough, h . 1. In that case, the piles will behave as individual piles. Thus, in practice, if h , 1, then Qg(u) 5 hS Qu and if h $ 1, then Qg(u) 5 S Qu There are several other equations like Eq. (11.119) for calculating the group efficiency of friction piles. Some of these are given in Table 11.17. It is important, however, to recognize that relationships such as Eq. (11.119) are simplistic and should not be used. In fact, in a group pile, the magnitude of fav depends on the location of the pile in the group (ex., Figure 11.38). 620 Chapter 11: Pile Foundations Table 11.17 Equations for Group Efficiency of Friction Piles Name Equation Converse–Labarre equation h512 B (n1 2 1)n2 1 (n2 2 1)n1 90n1n2 Ru where u(deg) 5 tan21 (D>d) Los Angeles Group Action equation h512 D 3n (n 2 1) pdn1n2 1 2 1 n2 (n1 2 1) 1 "2(n1 2 1) (n2 2 1) 4 Seiler–Keeney equation (Seiler and Keeney, 1944) h 5 b1 2 B n1 1 n2 2 2 0.3 11d RB Rr 1 n1 1 n2 2 1 n1 1 n2 7(d2 2 1) where d is in ft Sandy soil L = 18 D D = 250 mm Average skin friction, f (kN/m2) 50 Center pile Border pile 40 Corner pile 30 20 10 1.5D 3D 0 10 20 Settlement (mm) 30 Figure 11.38 Average skin friction (fav ) based on pile location (Based on Liu et al., 1985) 11.21 Ultimate Capacity of Group Piles in Saturated Clay 621 3 d d D Group efficiency, d 30° 2 d 35° 40° 1 45° 0 0 1 2 4 d D 6 8 Figure 11.39 Variation of efficiency of pile groups in sand (Based on Kishida and Meyerhof, 1965) Figure 11.39 shows the variation of the group efficiency h for a 3 3 3 group pile in sand (Kishida and Meyerhof, 1965). It can be seen that, for loose and medium sands, the magnitude of the group efficiency can be larger than unity. This is due primarily to the densification of sand surrounding the pile. 11.21 Ultimate Capacity of Group Piles in Saturated Clay Figure 11.40 shows a group pile in saturated clay. Using the figure, one can estimate the ultimate load-bearing capacity of group piles in the following manner: Step 1. Determine S Qu 5 n1n2 (Qp 1 Qs ). From Eq. (11.18), Qp 5 A p 39cu(p) 4 where cu(p) 5 undrained cohesion of the clay at the pile tip. Also, from Eq. (11.55), Qs 5 S apcu DL So, S Qu 5 n1n2 39A pcu(p) 1 S apcu DL4 (11.120) Step 2. Determine the ultimate capacity by assuming that the piles in the group act as a block with dimensions Lg 3 Bg 3 L. The skin resistance of the block is S pgcu DL 5 S 2(Lg 1 Bg )cu DL Calculate the point bearing capacity: A pqp 5 A pcu(p)N *c 5 (LgBg )cu(p)N *c 622 Chapter 11: Pile Foundations Qg(u) 2 (Lg Bg)cu L cu cu(1) L cu cu(2) cu cu(3) Lg Bg cu (p) N *c Bg Figure 11.40 Ultimate capacity of group piles in clay Lg Obtain the value of the bearing capacity factor N c* from Figure 11.41. Thus, the ultimate load is S Qu 5 LgBgcu(p)N c* 1 S 2(Lg 1 Bg )cu DL (11.121) Step 3. Compare the values obtained from Eqs. (11.120) and (11.121). The lower of the two values is Qg(u) . 9 Lg/Bg 1 8 2 3 N *c 7 6 5 4 0 1 2 3 L/Bg Figure 11.41 Variation of N c* with Lg>Bg and L>Bg 4 5 11.21 Ultimate Capacity of Group Piles in Saturated Clay 623 Example 11.18 The section of a 3 3 4 group pile in a layered saturated clay is shown in Figure 11.42. The piles are square in cross section (356 mm 3 356 mm). The center-to-center spacing, d, of the piles is 889 mm. Determine the allowable load-bearing capacity of the pile group. Use FS 5 4. Note that the groundwater table coincides with the ground surface. Solution From Eq. (11.120), SQu 5 n1n2 39A pcu(p) 1 a1pcu(1)L1 1 a2pcu(2)L24 From Figure 11.42, cu(1) 5 50.3 kN>m2 and cu(2) 5 85.1 kN>m2. For the top layer with cu(1) 5 50.3 kN>m2, cu(1) pa 5 50.3 5 0.503 100 From Table 11.10, a1 < 0.68. Similarly, cu(2) pa 5 85.1 < 0.85 100 a2 5 0.51 SQu 5 (3) (4) c (9) (0.356) 2 (85.1) 1 (0.68) (4 3 0.356) (50.3) (4.57) d 1 (0.51) (4 3 0.356) (85.1) (13.72) 5 14011 kN For piles acting as a group. Lg 5 (3) (0.889) 1 0.356 5 3.023 m Bg 5 (2) (0.889) 1 0.356 5 2.134 m G.W.T. Clay cu 50.3 kN/m2 sat 17.6 kN/m3 4.57 m Clay cu 85.1 kN/m2 sat 19.02 kN/m3 13.72 m 889 mm Figure 11.42 Group pile of layered saturated clay 624 Chapter 11: Pile Foundations Lg Bg 5 3.023 5 1.42 2.134 L 18.29 5 5 8.57 Bg 2.134 From Figure 11.41, N *c 5 8.75. From Eq. (11.121), SQu 5 LgBgcu(p)N*c 1 S2(Lg 1 Bg )cu DL 5 (3.023) (2.134) (85.1) (8.75) 1 (2) (3.023 12.134) 3(50.3) (4.57) 1 (85.1) (13.72) 4 5 19217 kN Hence, SQu 5 14,011 kN. SQall 5 11.22 14,011 14,011 5 < 3503 kN FS 4 ■ Elastic Settlement of Group Piles In general, the settlement of a group pile under a similar working load per pile increases with the width of the group (Bg ) and the center-to-center spacing of the piles (d). Several investigations relating to the settlement of group piles have been reported in the literature, with widely varying results. The simplest relation for the settlement of group piles was given by Vesic (1969), namely, sg(e) 5 Bg ÉD se (11.122) where sg(e) 5 elastic settlement of group piles Bg 5 width of group pile section D 5 width or diameter of each pile in the group se 5 elastic settlement of each pile at comparable working load (see Section 11.15) For group piles in sand and gravel, for elastic settlement, Meyerhof (1976) suggested the empirical relation sg(e) (mm) 5 0.96q"BgI N60 (11.123) 11.22 Elastic Settlement of Group Piles 625 where q 5 Qg> (LgBg ) (in kN>m2 ) (11.124) and Lg and Bg 5 length and width of the group pile section, respectively (m) N60 5 average standard penetration number within seat of settlement (<Bg deep below the tip of the piles) (11.125) I 5 influence factor 5 1 2 L>8Bg > 0.5 L 5 length of embedment of piles (m) Similarly, the group pile settlement is related to the cone penetration resistance by the formula Sg(e) 5 qBgI 2qc (11.126) where qc 5 average cone penetration resistance within the seat of settlement. (Note that, in Eq. (11.126), all quantities are expressed in consistent units.) Example 11.19 Consider a 3 3 4 group of prestressed concrete piles, each 21 m long, in a sand layer. The details of each pile and the sand are similar to that described in Example 11.10. The working load for the pile group is 6024 kN (3 3 4 3 Qall—where Qall 5 502 kN as in Example 11.10), and d>D 5 3. Estimate the elastic settlement of the pile group. Use Eq. (11.123). Solution se(g) 5 Bg ÉD se Bg 5 (3 2 1)d 1 2D 5 (2) (3D) 1 D 5 7D 5 (7) (0.356 m) 5 2.492 m 2 From Example 11.10, se 5 19.69 mm. Hence, se(g) 5 2.492 (19.69) 5 52.09 mm É 0.356 ■ 626 Chapter 11: Pile Foundations 11.23 Consolidation Settlement of Group Piles The consolidation settlement of a group pile in clay can be estimated by using the 2:1 stress distribution method. The calculation involves the following steps (see Figure 11.43): Step 1. Let the depth of embedment of the piles be L. The group is subjected to a total load of Qg . If the pile cap is below the original ground surface, Qg equals the total load of the superstructure on the piles, minus the effective weight of soil above the group piles removed by excavation. Step 2. Assume that the load Qg is transmitted to the soil beginning at a depth of 2L>3 from the top of the pile, as shown in the figure. The load Qg spreads out along two vertical to one horizontal line from this depth. Lines aar and bbr are the two 2:1 lines. Step 3. Calculate the increase in effective stress caused at the middle of each soil layer by the load Qg . The formula is Dsir 5 Qg (Bg 1 zi ) (Lg 1 zi ) Bg Lg Qg Clay layer 1 Groundwater table L a b 2 L 3 Clay layer 2 L1 z 2V:1H Clay layer 3 L2 2V:1H Clay layer 4 a L3 b Rock Figure 11.43 Consolidation settlement of group piles (11.127) 11.23 Consolidation Settlement of Group Piles 627 where Dsir 5 increase in effective stress at the middle of layer i Lg , Bg 5 length and width, respectively of the planned group piles zi 5 distance from z 5 0 to the middle of the clay layer i For example, in Figure 11.43, for layer 2, zi 5 L1>2; for layer 3, zi 5 L1 1 L2>2; and for layer 4, zi 5 L1 1 L2 1 L3>2. Note, however, that there will be no increase in stress in clay layer 1, because it is above the horizontal plane (z 5 0) from which the stress distribution to the soil starts. Step 4. Calculate the consolidation settlement of each layer caused by the increased stress. The formula is Dsc(i) 5 B De(i) 1 1 eo(i) RHi (11.128) where Dsc(i) 5 consolidation settlement of layer i De(i) 5 change of void ratio caused by the increase in stress in layer i eo(i) 5 initial void ratio of layer i (before construction) Hi 5 thickness of layer i (Note: In Figure 11.43, for layer 2, Hi 5 L1 ; for layer 3, Hi 5 L2 ; and for layer 4, Hi 5 L3 .) Relationships involving De(i) are given in Chapter 1. Step 5. The total consolidation settlement of the group piles is then Dsc(g) 5 SDsc(i) (11.129) Note that the consolidation settlement of piles may be initiated by fills placed nearby, adjacent floor loads, or the lowering of water tables. Example 11.20 A group pile in clay is shown in Figure 11.44. Determine the consolidation settlement of the piles. All clays are normally consolidated. Solution Because the lengths of the piles are 15 m each, the stress distribution starts at a depth of 10 m below the top of the pile. We are given that Qg 5 2000 kN. Calculation of Settlement of Clay Layer 1 For normally consolidated clays, Dsc(1) 5 c Ds(1) r 5 (Cc(1)H1 ) 1 1 eo(1) d log c so(1) r 1 Ds(1) r Qg (Lg 1 z1 ) (Bg 1 z1 ) so(1) r 5 d 2000 5 51.6 kN>m2 (3.3 1 3.5) (2.2 1 3.5) 628 Chapter 11: Pile Foundations Qg 2000 kN Sand 16.2 kN/m3 Groundwater table 10 m 15 m 2m 1m Clay 9m Group pile 16 m 7m z 2V:1H o(1) (1) Clay o(2) (2) 4m o(3), (3) 2m Clay 2V:1H Rock sat 18.0 kN/m3 eo 0.82 Cc 0.3 sat 18.9 kN/m3 eo 0.7 Cc 0.2 sat 19 kN/m3 eo 0.75 Cc 0.25 Pile group: Lg 3.3 m; Bg 2.2 m (not to scale) Figure 11.44 Consolidation settlement of a pile group and so(1) r 5 2(16.2) 1 12.5(18.0 2 9.81) 5 134.8 kN>m2 So Dsc(1) 5 (0.3) (7) 134.8 1 51.6 log c d 5 0.1624 m 5 162.4 mm 1 1 0.82 134.8 Settlement of Layer 2 As with layer 1, Dsc(2) 5 Cc(2)H2 1 1 eo(2) log c so(2) r 1 Ds(2) so(2) r d ss(2) r 5 2(16.2) 1 16(18.0 2 9.81) 1 2(18.9 2 9.81) 5 181.62 kN>m2 and Ds(2) r 5 2000 5 14.52 kN>m2 (3.3 1 9) (2.2 1 9) Hence, Dsc(2) 5 (0.2) (4) 181.62 1 14.52 log c d 5 0.0157 m 5 15.7 mm 1 1 0.7 181.62 Problems 629 Settlement of Layer 3 Continuing analogously, we have so(3) r 5 181.62 1 2(18.9 2 9.81) 1 1(19 2 9.81) 5 208.99 kN>m2 Ds(3) r 5 2000 5 9.2 kN>m2 (3.3 1 12) (2.2 1 12) Dsc(3) 5 (0.25) (2) 208.99 1 9.2 log a b 5 0.0054 m 5 5.4 mm 1 1 0.75 208.99 Hence, the total settlement is Dsc(g) 5 162.4 1 15.7 1 5.4 5 183.5 mm 11.24 ■ Piles in Rock For point bearing piles resting on rock, most building codes specify that Qg(u) 5 S Qu , provided that the minimum center-to-center spacing of the piles is D 1 300 mm. For Hpiles and piles with square cross sections, the magnitude of D is equal to the diagonal dimension of the cross section of the pile. Problems 11.1 A 12 m long concrete pile is shown in Figure P11.1. Estimate the ultimate point load Qp by a. Meyerhof’s method b. Vesic’s method c. Coyle and Castello’s method Use m 5 600 in Eq. (11.26). 11.2 Refer to the pile shown in Figure P11.1. Estimate the side resistance Qs by a. Using Eqs. (11.40) through (11.42). Use K 5 1.3 and dr 5 0.8fr b. Coyle and Castello’s method [Eq. (11.44)] 11.3 Based on the results of Problems 11.1 and 11.2, recommend an allowable load for the pile. Use FS 5 4. 11.4 A driven closed-ended pile, circular in cross section, is shown in Figure P11.4. Calculate the following. a. The ultimate point load using Meyerhof’s procedure. b. The ultimate point load using Vesic’s procedure. Take Irr 5 50. c. An approximate ultimate point load on the basis of parts (a) and (b). d. The ultimate frictional resistance Qs. [Use Eqs. (11.40) through (11.42), and take K 5 1.4 and dr 5 0.6fr.] e. The allowable load of the pile (use FS 5 4). 11.5 Following is the variation of N60 with depth in a granular soil deposit. A concrete pile 9 m long (0.305 m 3 0.305 m in cross section) is driven into the sand and fully embedded in the sand. 630 Chapter 11: Pile Foundations Concrete pile 356 mm 356 mm Loose sand 1 30º 17.5 kN/m3 12 m Dense sand 2 42º 18.5 kN/m3 3.05 m Groundwater table 3.05 m Figure P11.1 15.72 kN/m3 32º c 0 sat 18.24 kN/m3 32° c 0 sat 19.24 kN/m3 40º c 0 15.24 m Figure P11.4 15 in. Depth (m) N60 1.5 3.0 4.5 6.0 7.5 9.0 10.5 11.0 12.5 14.0 4 8 7 5 16 18 21 24 20 19 Problems 631 Silty clay sat 18.55 kN/m3 cu 35 kN/m2 6.1 m Groundwater table Silty clay sat 19.24 kN/m3 cu 75 kN/m2 12.2 m 406 mm 11.6 11.7 11.8 11.9 11.10 11.11 11.12 Figure P11.10 Estimate the allowable load-carrying capacity of the pile (Qall). Use FS 5 4 and Meyerhof’s equations [Eqs. (11.37) and (11.45)]. Solve Problem 11.5 using the equation of Briaud et al. [Eqs. (11.38) and (11.47)]. A concrete pile 15.24 m long having a cross section of 406 mm 3 406 mm is fully embedded in a saturated clay layer for which gsat 5 19.02 kN>m3, f 5 0, and cu 5 76.7 kN>m2. Determine the allowable load that the pile can carry. (Let FS 5 3.) Use the a method to estimate the skin friction and Veric’s method for point load estimation. Redo Problem 11.7 using the l method for estimating the skin friction and Meyerhof’s method for the point load estimation. A concrete pile 15 m long having a cross section of 0.38 m 3 0.38 m is fully embedded in a saturated clay layer. For the clay, given: gsat 5 18 kN>m3, f 5 0, and cu 5 80 kN>m2. Determine the allowable load that the pile can carry (FS 5 3). Use the l method to estimate the skin resistance. A concrete pile 406 mm 3 406 mm in cross section is shown in Figure P11.10. Calculate the ultimate skin friction resistance by using the a. a method b. l method c. b method Use fRr 5 20° for all clays, which are normally consolidated. A steel pile (H-section; HP 360 3 152; see Table 11.1) is driven into a layer of sandstone. The length of the pile is 18.9 m. Following are the properties of the sandstone: unconfined compression strength 5 qu(lab) 5 78.7 MN/m2 and angle of friction 5 36°. Using a factor of safety of 3, estimate the allowable point load that can be carried by the pile. Use 3qu(design) 5 qu(lab)>54. A concrete pile is 18 m long and has a cross section of 0.406 m 3 0.406 m. The pile is embedded in a sand having g 5 16 kN>m3 and fr 5 37°. The allowable 632 Chapter 11: Pile Foundations 11.13 11.14 11.15 11.16 11.17 11.18 11.19 11.20 11.21 d working load is 900 kN. If 600 kN are contributed by the frictional resistance and 300 kN are from the point load, determine the elastic settlement of the pile. Given: Ep 5 2.1 3 106 kN/m.2, Es 5 30 3 103 kN/m.2, ms 5 0.38, and j 5 0.57 [Eq. (11.73)]. Solve Problem 11.12 with the following: length of pile 5 15 m, pile cross section 5 0.305 m 3 0.305 m, allowable working load 5 338 kN, contribution of frictional resistance to working load 5 280 kN, Ep 5 21 3 106 kN>m2, Es 5 30,000 kN>m2, ms 5 0.3, and j 5 0.62 [Eq. (11.73)]. A 30-m long concrete pile is 305 mm 3 305 mm in cross section and is fully embedded in a sand deposit. If nh 5 9200 kN>m2, the moment at ground level, Mg 5 0, the allowable displacement of pile head 5 12 mm; Ep 5 21 3 106 kN>m2; and FY (pile) 5 21,000 kN>m2, calculate the allowable lateral load, Qg, at the ground level. Use the elastic solution method. Solve Problem 11.14 by Brom’s method. Assume that the pile is flexible and free headed. Let the soil unit weight, g 5 16 kN>m3; the soil friction angle, fr 5 30°; and the yield stress of the pile material, FY 5 21 MN>m2. A steel H-pile (section HP 330 3 149) is driven by a hammer. The maximum rated hammer energy is 54.23 kN-m, the weight of the ram is 53.4 kN, and the length of the pile is 27.44 m. Also, we have coefficient of restitution 5 0.35, weight of the pile cap 5 10.7 kN, hammer efficiency 5 0.85, number of blows for the last inch of penetration 5 10, and Ep 5 207 3 106 kN>m2. Estimate the pile capacity using Eq. (11.106). Take FS 5 6. Solve Problem 11.16 using the modified EN formula. (See Table 11.16). Use FS 5 4. Solve Problem 11.16 using the Danish formula (See Table 11.16). Use FS 5 3. Figure 11.35a shows a pile. Let L 5 20 m, D (pile diameter) 5 450 mm, Hf 5 4 m, gfill 5 17.5 kN>m3, and frfill 5 25°. Determine the total downward drag force on the pile. Assume that the fill is located above the water table and r . that d r 5 0.5ffill Redo Problem 11.19 assuming that the water table coincides with the top of the fill and that gsat(fill) 5 19.8 kN>m3 . If the other quantities remain the same, what would be the downward drag force on the pile? Assume dr 5 0.5ffill r . 3 Refer to Figure 11.35b. Let L 5 15.24 m, gfill 5 17.29 kN>m , gsat(clay) 5 19.49 kN>m3, fclay r 5 20°, Hf 5 3.05 m, and D (pile diameter) 5 406 mm. The water table coincides with the top of the clay layer. Determine the total downward drag force on the pile. Assume dr 5 0.6fclay r . Figure p11.23 Problems 633 Clay cu 25 kN/m2 5m Clay cu 45 kN/m2 6m 6m Clay cu 60 kN/m2 1m Figure P11.25 11.22 A concrete pile measuring 0.406 m 3 0.406 m in cross section is 18.3 m long. It is fully embedded in a layer of sand. The following is an approximation of the mechanical cone penetration resistance (qc ) and the friction ratio (Fr ) for the sand layer. Estimate the allowable bearing capacity of the pile. Use FS 5 4. Depth below ground surface (m) 0–6.1 6.1–13.7 13.7–19.8 qc (kN , m2 ) Fr (%) 2803 3747 8055 2.3 2.7 2.8 11.23 The plan of a group pile is shown in Figure P11.23. Assume that the piles are embedded in a saturated homogeneous clay having a cu 5 86 kN>m2. Given: diameter of piles (D) 5 316 mm, center-to-center spacing of piles 5 600 mm, and length of piles 5 20 m. Find the allowable load-carrying capacity of the pile group. Use FS 5 3. 11.24 Redo Problem 11.23 with the following: center-to-center spacing of piles 5 762 mm, length of piles 5 13.7 m, D 5 305 mm, cu 5 41.2 kN>m2, gsat 5 19.24 kN>m3, and FS 5 3. 11.25 The section of a 4 3 4 group pile in a layered saturated clay is shown in Figure P11.25. The piles are square in cross section (356 mm 3 356 mm). The center-to-center spacing (d) of the piles is 1 m. Determine the allowable load-bearing capacity of the pile group. Use FS 5 3. 11.26 Figure P11.26 shows a group pile in clay. Determine the consolidation settlement of the group. Use the 2:1 method of estimate the average effective stress in the clay layers. 634 Chapter 11: Pile Foundations 1335 kN 3m Groundwater table Sand = 15.72 kN/m3 Sand sat = 18.55 kN/m3 3m 2.75 m 2.75 m Group plan 18 m Normally consolidated clay sat = 19.18 kN/m3 eo = 0.8 Cc = 0.8 5m Normally consolidated clay sat = 18.08 kN/m3 eo = 1.0 Cc = 0.31 3m Normally consolidated clay sat = 19.5 kN/m3 eo = 0.7 Cc = 0.26 15 m Rock Figure P11.26 References AMERICAN SOCIETY OF CIVIL ENGINEERS (1959). “Timber Piles and Construction Timbers,” Manual of Practice, No. 17, American Society of Civil Engineers, New York. AMERICAN SOCIETY OF CIVIL ENGINEERS (1993). Design of Pile Foundations (Technical Engineering and Design Guides as Adapted from the U.S. Army Corps of Engineers, No. 1), American Society of Civil Engineers, New York. BALDI, G., BELLOTTI, R., GHIONNA, V., JAMIOLKOWSKI, M. and PASQUALINI, E. (1981). “Cone Resistance in Dry N.C. and O.C. Sands, Cone Penetration Testing and Experience,” Proceedings, ASCE Specialty Conference, St. Louis, pp. 145–177. BJERRUM, L., JOHANNESSEN, I. J., and EIDE, O. (1969). “Reduction of Skin Friction on Steel Piles to Rock,” Proceedings, Seventh International Conference on Soil Mechanics and Foundation Engineering, Mexico City, Vol. 2, pp. 27–34. References 635 BOWLES, J. E. (1982). Foundation Analysis and Design, McGraw-Hill, New York. BOWLES, J. E. (1996). Foundation Analysis and Design, McGraw-Hill, New York. BRIAUD, J. L., TUCKER, L., LYTTON, R. L., and COYLE, H. M. (1985). Behavior of Piles and Pile Groups, Report No. FHWA>RD-83>038, Federal Highway Administration, Washington, DC. BROMS, B. B. (1965). “Design of Laterally Loaded Piles,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 91, No. SM3, pp. 79 – 99. CHEN, Y. J., and KULHAWY, F. H. (1994). “Case History Evaluation of the Behavior of Drilled Shafts under Axial and Lateral Loading,” Final Report, Project 1493-04, EPRI TR-104601, Geotechnical Group, Cornell University, Ithaca, NY, December. COYLE, H. M., and CASTELLO, R. R. (1981). “New Design Correlations for Piles in Sand,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 107, No. GT7, pp. 965–986. DAVISSON, M. T. (1973). “High Capacity Piles” in Innovations in Foundation Construction, Proceedings of a Lecture Series, Illinois Section, American Society of Civil Engineers, Chicago. DAVISSON, M. T. (1970). “BRD Vibratory Driving Formula,” Foundation Facts, Vol. VI, No. 1, pp. 9–11. DAVISSON, M. T., and GILL, H. L. (1963). “Laterally Loaded Piles in a Layered Soil System,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 89, No. SM3, pp. 63 –94. FENG, Z., and DESCHAMPS, R. J. (2000). “A Study of the Factors Influencing the Penetration and Capacity of Vibratory Driven Piles,” Soils and Foundations, Vol. 40, No. 3, pp. 43–54. GOODMAN, R. E. (1980). Introduction to Rock Mechanics, Wiley, New York. GUANG-YU, Z. (1988). “Wave Equation Applications for Piles in Soft Ground,” Proceedings, Third International Conference on the Application of Stress-Wave Theory to Piles (B. H. Fellenius, ed.), Ottawa, Ontario, Canada, pp. 831–836. JANBU, N. (1953). An Energy Analysis of Pile Driving with the Use of Dimensionless Parameters, Norwegian Geotechnical Institute, Oslo, Publication No. 3. KISHIDA, H., and MEYERHOF, G. G. (1965). “Bearing Capacity of Pile Groups under Eccentric Loads in Sand,” Proceedings, Sixth International Conference on Soil Mechanics and Foundation Engineering, Montreal, Vol. 2, pp. 270–274. LIU, J. L., YUAN, Z. I., and ZHANG, K. P. (1985). “Cap-Pile-Soil Interaction of Bored Pile Groups,” Proceedings, Eleventh International Conference on Soil Mechanics and Foundation Engineering, San Francisco, Vol. 3, pp. 1433–1436. MANSUR, C. I., and HUNTER, A. H. (1970). “Pile Tests—Arkansas River Project,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 96, No. SM6, pp. 1545–1582. MATLOCK, H., and REESE, L. C. (1960). “Generalized Solution for Laterally Loaded Piles,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 86, No. SM5, Part I, pp. 63 –91. MEYERHOF, G. G. (1976). “Bearing Capacity and Settlement of Pile Foundations,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 102, No. GT3, pp. 197–228. NOTTINGHAM, L. C., and SCHMERTMANN, J. H. (1975). An Investigation of Pile Capacity Design Procedures, Research Report No. D629, Department of Civil Engineering, University of Florida, Gainesville, FL. OLSON, R. E., and FLAATE, K. S. (1967). “Pile Driving Formulas for Friction Piles in Sand,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 93, No. SM6, pp. 279 –296. O’NEILL, M. W., and REESE, L. C. (1999). Drilled Shafts: Construction Procedure and Design Methods, FHWA Report No. IF-99-025. SCHMERTMANN, J. H. (1978). Guidelines for Cone Penetration Test: Performance and Design, Report FHWA-TS-78-209, Federal Highway Administration, Washington, DC. SEILER, J. F., and KEENEY, W. D. (1944). “The Efficiency of Piles in Groups,” Wood Preserving News, Vol. 22, No. 11 (November). 636 Chapter 11: Pile Foundations SKOV, R., and DENVER, H. (1988). “Time Dependence of Bearing Capacity of Piles,” Proceedings, Third International Conference on Application of Stress Wave Theory to Piles, Ottawa, Canada, pp. 879 –889. SLADEN, J. A. (1992). “The Adhesion Factor: Applications and Limitations,” Canadian Geotechnical Journal, Vol. 29, No. 2, pp. 323–326. SVINKIN, M. (1996). Discussion on “Setup and Relaxation in Glacial Sand,” Journal of Geo-technical Engineering, ASCE, Vol. 22, pp. 319–321. TERZAGHI, K., PECK, R. B., and MESRI, G. (1996). Soil Mechanics in Engineering Practice, John Wiley, NY. VESIC, A. S. (1961). “Bending of Beams Resting on Isotropic Elastic Solids,” Journal of the Engineering Mechanics Division, American Society of Civil Engineers, Vol. 87, No. EM2, pp. 35–53. VESIC, A. S. (1969). Experiments with Instrumented Pile Groups in Sand, American Society for Testing and Materials, Special Technical Publication No. 444, pp. 177–222. VESIC, A. S. (1970). “Tests on Instrumental Piles—Ogeechee River Site,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 96, No. SM2, pp. 561–584. VESIC, A. S. (1977). Design of Pile Foundations, National Cooperative Highway Research Program Synthesis of Practice No. 42, Transportation Research Board, Washington, DC. VIJAYVERGIYA, V. N., and FOCHT, J. A., JR. (1972). A New Way to Predict Capacity of Piles in Clay, Offshore Technology Conference Paper 1718, Fourth Offshore Technology Conference, Houston. WONG, K. S., and TEH, C. I. (1995). “Negative Skin Friction on Piles in Layered Soil Deposit,” Journal of Geotechnical and Geoenvironmental Engineering, American Society of Civil Engineers, Vol. 121, No. 6, pp. 457– 465. WOODWARD, R. J., GARDNER, W. S., and GREER, D. M. (1972). Drilled Pier Foundations, McGraw-Hill, New York. 12 12.1 Drilled-Shaft Foundations Introduction The terms caisson, pier, drilled shaft, and drilled pier are often used interchangeably in foundation engineering; all refer to a cast-in-place pile generally having a diameter of about 750 mm or more, with or without steel reinforcement and with or without an enlarged bottom. Sometimes the diameter can be as small as 305 mm. To avoid confusion, we use the term drilled shaft for a hole drilled or excavated to the bottom of a structure’s foundation and then filled with concrete. Depending on the soil conditions, casings may be used to prevent the soil around the hole from caving in during construction. The diameter of the shaft is usually large enough for a person to enter for inspection. The use of drilled-shaft foundations has several advantages: 1. A single drilled shaft may be used instead of a group of piles and the pile cap. 2. Constructing drilled shafts in deposits of dense sand and gravel is easier than driving piles. 3. Drilled shafts may be constructed before grading operations are completed. 4. When piles are driven by a hammer, the ground vibration may cause damage to nearby structures. The use of drilled shafts avoids this problem. 5. Piles driven into clay soils may produce ground heaving and cause previously driven piles to move laterally. This does not occur during the construction of drilled shafts. 6. There is no hammer noise during the construction of drilled shafts; there is during pile driving. 7. Because the base of a drilled shaft can be enlarged, it provides great resistance to the uplifting load. 8. The surface over which the base of the drilled shaft is constructed can be visually inspected. 9. The construction of drilled shafts generally utilizes mobile equipment, which, under proper soil conditions, may prove to be more economical than methods of constructing pile foundations. 10. Drilled shafts have high resistance to lateral loads. 637 638 Chapter 12: Drilled-Shaft Foundations There are also a couple of drawbacks to the use of drilled-shaft construction. For one thing, the concreting operation may be delayed by bad weather and always needs close supervision. For another, as in the case of braced cuts, deep excavations for drilled shafts may induce substantial ground loss and damage to nearby structures. 12.2 Types of Drilled Shafts Drilled shafts are classified according to the ways in which they are designed to transfer the structural load to the substratum. Figure 12.1a shows a drilled straight shaft. It extends through the upper layer(s) of poor soil, and its tip rests on a strong load-bearing soil layer or rock. The shaft can be cased with steel shell or pipe when required (as it is with cased, cast-in-place concrete piles; see Figure 11.4). For such shafts, the resistance to the applied load may develop from end bearing and also from side friction at the shaft perimeter and soil interface. A belled shaft (see Figures 12.1b and c) consists of a straight shaft with a bell at the bottom, which rests on good bearing soil. The bell can be constructed in the shape of a dome (see Figure 12.1b), or it can be angled (see Figure 12.1c). For angled bells, the underreaming tools that are commercially available can make 30 to 45° angles with the vertical. For the majority of drilled shafts constructed in the United States, the entire load-carrying capacity is assigned to the end bearing only. However, under certain circumstances, the end-bearing capacity and the side friction are taken into account. In Europe, both the side frictional resistance and the end-bearing capacity are always taken into account. Straight shafts can also be extended into an underlying rock layer. (See Figure 12.1d.) In the calculation of the load-bearing capacity of such shafts, the end bearing and the shear stress developed along the shaft perimeter and rock interface can be taken into account. Soft soil Soft soil 45˚ or 30˚ Rock or hard soil (a) Good bearing soil (b) Good bearing soil (c) Rock 0.15 to 0.3 m (d) Figure 12.1 Types of drilled shaft: (a) straight shaft; (b) and (c) belled shaft; (d) straight shaft socketed into rock 12.3 Construction Procedures 12.3 639 Construction Procedures The most common construction procedure used in the United States involves rotary drilling. There are three major types of construction methods: the dry method, the casing method, and the wet method. Dry Method of Construction This method is employed in soils and rocks that are above the water table and that will not cave in when the hole is drilled to its full depth. The sequence of construction, shown in Figure 12.2, is as follows: Step 1. The excavation is completed (and belled if desired), using proper drilling tools, and the spoils from the hole are deposited nearby. (See Figure 12.2a.) Step 2. Concrete is then poured into the cylindrical hole. (See Figure 12.2b.) Step 3. If desired, a rebar cage is placed in the upper portion of the shaft. (See Figure 12.2c.) Step 4. Concreting is then completed, and the drilled shaft will be as shown in Figure 12.2d. Casing Method of Construction This method is used in soils or rocks in which caving or excessive deformation is likely to occur when the borehole is excavated. The sequence of construction is shown in Figure 12.3 and may be explained as follows: Step 1. The excavation procedure is initiated as in the case of the dry method of construction. (See Figure 12.3a.) Step 2. When the caving soil is encountered, bentonite slurry is introduced into the borehole. (See Figure 12.3b.) Drilling is continued until the excavation goes past the caving soil and a layer of impermeable soil or rock is encountered. Step 3. A casing is then introduced into the hole. (See Figure 12.3c.) Step 4. The slurry is bailed out of the casing with a submersible pump. (See Figure 12.3d.) Step 5. A smaller drill that can pass through the casing is introduced into the hole, and excavation continues. (See Figure 12.3e.) Step 6. If needed, the base of the excavated hole can then be enlarged, using an underreamer. (See Figure 12.3f.) Step 7. If reinforcing steel is needed, the rebar cage needs to extend the full length of the excavation. Concrete is then poured into the excavation and the casing is gradually pulled out. (See Figure 12.3g.) Step 8. Figure 12.3h shows the completed drilled shaft. Wet Method of Construction This method is sometimes referred to as the slurry displacement method. Slurry is used to keep the borehole open during the entire depth of excavation. (See Figure 12.4.) Following are the steps involved in the wet method of construction: 640 Chapter 12: Drilled-Shaft Foundations Surface casing, if required Competent, noncaving soil Drop chute (a) Competent, noncaving soil (b) Competent, noncaving soil Competent, noncaving soil (c) (d) Figure 12.2 Dry method of construction: (a) initiating drilling; (b) starting concrete pour; (c) placing rebar cage; (d) completed shaft (After O’Neill and Reese, 1999) 12.3 Construction Procedures Drilling slurry Cohesive soil Cohesive soil Caving soil Caving soil Cohesive soil Cohesive soil (a) (b) Cohesive soil Cohesive soil Caving soil Caving soil Cohesive soil Cohesive soil (c) (d) Figure 12.3 Casing method of construction: (a) initiating drilling; (b) drilling with slurry; (c) introducing casing; (d) casing is sealed and slurry is being removed from interior of casing; (e) drilling below casing; (f) underreaming; (g) removing casing; (h) completed shaft (After O’Neill and Reese, 1999) 641 642 Chapter 12: Drilled-Shaft Foundations Competent soil Competent soil Caving soil Caving soil Competent soil Competent soil (e) (f) Level of fluid concrete Drifting fluid forced from space between casing and soil Competent soil Competent soil Caving soil Caving soil Competent soil Competent soil (g) Figure 12.3 (Continued) (h) 12.3 Construction Procedures Drilling slurry Cohesive soil Caving soil 643 Cohesive soil Caving soil (b) (a) Cohesive soil Cohesive soil Sump Caving soil Caving soil (c) (d) Figure 12.4 Slurry method of construction: (a) drilling to full depth with slurry; (b) placing rebar cage; (c) placing concrete; (d) completed shaft (After O’Neill and Reese, 1999) Step 1. Excavation continues to full depth with slurry. (See Figure 12.4a.) Step 2. If reinforcement is required, the rebar cage is placed in the slurry. (See Figure 12.4b.) Step 3. Concrete that will displace the volume of slurry is then placed in the drill hole. (See Figure 12.4c.) Step 4. Figure 12.4d shows the completed drilled shaft. Figure 12.5 shows a drilled shaft under construction using the dry method. The construction of a drilled shaft using the wet method is shown in Figure 12.6. A typical auger, a reinforcement cage, and a typical clean-out bucket are shown in Figure 12.7. 644 Chapter 12: Drilled-Shaft Foundations Figure 12.5 Drilled shaft construction using the dry method (Courtesy of Sanjeev Kumar, Southern Illinois University, Carbondale, Illinois) Figure 12.6 Drilled shaft construction using wet method (Courtesy of Khaled Sobhan, Florida Atlantic Univetsity, Boca Raton, Florida) 12.4 Other Design Considerations (a) 645 (b) (c) Figure 12.7 Drilled shaft construction: (a) A typical auger; (b) a reinforcement cage; (c) a cleanout bucket (Courtesy of Khaled Sobhan, Florida Atlantic Univetsity, Boca Raton, Florida) 12.4 Other Design Considerations For the design of ordinary drilled shafts without casings, a minimum amount of vertical steel reinforcement is always desirable. Minimum reinforcement is 1% of the gross cross-sectional area of the shaft. For drilled shafts with nominal reinforcement, most building codes suggest using a design concrete strength, fc , on the order of fcr >4. Thus, the minimum shaft diameter becomes fc 5 0.25fcr 5 Qw Qw 5 p 2 A gs Ds 4 646 Chapter 12: Drilled-Shaft Foundations or Ds 5 Qw p ¢ ≤ (0.25)fcr ç 4 Qu Å fcr 5 2.257 (12.1) where Ds 5 diameter of the shaft fcr 5 28-day concrete strength Qw 5 working load of the drilled shaft A gs 5 gross cross-sectional area of the shaft If drilled shafts are likely to be subjected to tensile loads, reinforcement should be continued for the entire length of the shaft. Concrete Mix Design The concrete mix design for drilled shafts is not much different from that for any other concrete structure. When a reinforcing cage is used, consideration should be given to the ability of the concrete to flow through the reinforcement. In most cases, a concrete slump of about 15.0 mm (6 in.) is considered satisfactory. Also, the maximum size of the aggregate should be limited to about 20 mm (0.75 in.) 12.5 Load Transfer Mechanism The load transfer mechanism from drilled shafts to soil is similar to that of piles, as described in Section 11.5. Figure 12.8 shows the load test results of a drilled shaft, conducted in a clay soil by Reese et al. (1976). The shaft (Figure 12.8a) had a diameter of 762 mm and a depth of penetration of 6.94 m. Figure 12.8b shows the load-settlement curves. It can be seen that the total load carried by the drilled shaft was 1246 kN. The load carried by side resistance was about 800 kN, and the rest was carried by point bearing. It is interesting to note that, with a downward movement of about 6 mm, full side resistance was mobilized. However, about 25 mm of downward movement was required for mobilization of full point resistance. This situation is similar to that observed in the case of piles. Figure 12.8c shows the average load-distribution curves for different stages of the loading. 12.6 Estimation of Load-Bearing Capacity The ultimate load-bearing capacity of a drilled shaft (see Figure 12.9) is Qu 5 Qp 1 Qs (12.2) 12.6 Estimation of Load-Bearing Capacity 647 Steel loading plate 0.762 m 1200 Load (kN) 762 mm 6.94 m Total 800 Sides 400 0 114 mm Base 0 Bottom hole load cell (a) 0 0 6 12 18 24 Mean settlement (mm) 30 (b) 400 Load (kN) 800 1200 1400 Depth (m) 1.5 3.0 4.5 6.0 7.5 (c) Figure 12.8 Load test results of Reese et al. (1976) on a drilled shaft: (a) dimensions of the shaft; (b) plot of base, sides, and total load with mean settlement; (c) plot of loaddistribution curve with depth where Qu 5 ultimate load Qp 5 ultimate load-carrying capacity at the base Qs 5 frictional (skin) resistance The ultimate base load Qp can be expressed in a manner similar to the way it is expressed in the case of shallow foundations [Eq. (3.19)], or 1 Qp 5 A p acrNcFcsFcdFcc 1 qrNqFqsFqdFqc 1 grNgFgsFgdFgc b (12.3) 2 648 Chapter 12: Drilled-Shaft Foundations Qu Qu Qs Qs z z L1 DS Db Qp (a) L Soil c L L1 Db Ds Qp Soil c (b) Figure 12.9 Ultimate bearing capacity of drilled shafts: (a) with bell and (b) straight shaft where cr 5 cohesion Nc , Nq , Ng 5 bearing capacity factors Fcs , Fqs , Fgs 5 shape factors Fcd , Fqd , Fgd 5 depth factors Fcc , Fqc , Fgc 5 compressibility factors gr 5 effective unit weight of soil at the base of the shaft qr 5 effective vertical stress at the base of the shaft p A p 5 area of the base 5 D2b 4 In most instances, the last term (the one containing Ng) is neglected, except in the case of a relatively short drilled shaft. With this assumption, we can write Qu 5 A p (c rNcFcsFcdFcc 1 qNqFqsFqdFqc ) 1 Qs (12.4) The procedure to estimate the ultimate capacity of drilled shafts in granular and cohesive soil is described in the following sections. 12.7 Drilled Shafts in Granular Soil: Load-Bearing Capacity Estimation of Qp For a drilled shaft with its base located on a granular soil (that is, c r 5 0), the net ultimate load-carrying capacity at the base can be obtained from Eq. (12.4) as Qp(net) 5 A p 3q r (Nq 2 1)FqsFqdFqc4 (12.5) 649 12.7 Drilled Shafts in Granular Soil: Load-Bearing Capacity The bearing capacity factor, Nq, for various soil friction angles (fr ) can be taken from Table 3.3. It is also given in Table 12.1. Also, Fqs 5 1 1 tan fr (12.6) Fqd 5 1 1 C tan21 a L b Db (')'* radian (12.7) C 5 2 tan fr (1 2 sin fr ) 2 (12.8) The variations of Fqs and C with fr are given in Table 12.1. According to Chen and Kulhawy (1994), Fqc can be calculated in the following manner. Step 1. Calculate the critical rigidity index as Icr 5 0.5 exp c2.85 cota45 2 fr bd 2 (12.9) where Icr 5 critical rigidity index (see Table 12.1). Table 12.1 Variation of Nq, Fqs, C, Icr, ms, and n with fr Soil friction angle, f9 (deg) Nq (Table 3.3) Fqs [Eq. (12.6)] C [Eq. (12.8)] Icr [Eq. (12.9)] ms [Eq. (12.13)] n [Eq. (12.15)] 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 10.66 11.85 13.20 14.72 16.44 18.40 20.63 23.18 26.09 29.44 33.30 37.75 42.92 48.93 55.96 64.20 73.90 85.38 99.02 115.31 134.88 1.466 1.488 1.510 1.532 1.554 1.577 1.601 1.625 1.649 1.675 1.700 1.727 1.754 1.781 1.810 1.839 1.869 1.900 1.933 1.966 2.000 0.311 0.308 0.304 0.299 0.294 0.289 0.283 0.276 0.269 0.262 0.255 0.247 0.239 0.231 0.223 0.214 0.206 0.197 0.189 0.180 0.172 43.84 47.84 52.33 57.40 63.13 69.63 77.03 85.49 95.19 106.37 119.30 134.33 151.88 172.47 196.76 225.59 259.98 301.29 351.22 412.00 486.56 0.100 0.115 0.130 0.145 0.160 0.175 0.190 0.205 0.220 0.235 0.250 0.265 0.280 0.295 0.310 0.325 0.340 0.355 0.370 0.385 0.400 0.00500 0.00475 0.00450 0.00425 0.00400 0.00375 0.00350 0.00325 0.00300 0.00275 0.00250 0.00225 0.00200 0.00175 0.00150 0.00125 0.00100 0.00075 0.00050 0.00025 0.00000 650 Chapter 12: Drilled-Shaft Foundations Step 2. Calculate the reduced rigidity index as Irr 5 Ir 1 1 Ir D (12.10) where Ir 5 soil rididity index 5 Es 2(1 1 ms )q r tan fr (12.11) in which Es 5 drained modulus of elasticity of soil 5 mpa pa 5 atmospheric pressure( < 100 kN>m2 ) (12.12) 100 to 200 (loose soil) m 5 c200 to 500 (medium dense soil) 500 to 1000(dense soil) ms 5 Poisson’s ratio of soil 5 0.1 1 0.3a fr 2 25 b 20 (for 25° # f r # 45°) (see Table12.1) D5n n 5 0.005a1 2 qr pa (12.13) (12.14) fr 2 25 b (see Table 12.1) 20 (12.15) Step 3. If Irr $ Icr, then Fqc 5 1 (12.16) However, if Irr , Icr, then Fqc 5 exp e (23.8 tan fr ) 1 c (3.07 sin fr ) ( log 10 2Irr ) df 1 1 sin fr (12.17) The magnitude of Qp(net) also can be reasonably estimated from a relationship based on the analysis of Berezantzev et al. (1961) that can be expressed as Qp(net) 5 A pq r (vN *q 2 1) (12.18) where N *q 5 bearing capacity factor 5 0.21e0.17fr (See Table 12.2) v 5 correction factor 5 f(L>Db ) (12.19) In Eq. (12.19), f r is in degrees. The variation of v with L>Db is given in Figure 12.10. 12.7 Drilled Shafts in Granular Soil: Load-Bearing Capacity 651 Table 12.2 Variation of N q* with fr [Eq. (12.19)] fr (deg) N*q 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 14.72 17.45 20.68 24.52 29.06 34.44 40.83 48.39 57.36 67.99 80.59 95.52 113.22 134.20 159.07 188.55 223.49 264.90 313.99 372.17 441.14 1.0 0.9 L/Db 0.8 5 0.7 10 15 0.6 20 25 0.5 0.4 26 28 30 32 34 36 Soil friction angle, (deg) 38 40 Figure 12.10 Variation of v with fr and L>Db 652 Chapter 12: Drilled-Shaft Foundations Estimation of Qs The frictional resistance at ultimate load, Qs , developed in a drilled shaft may be calculated as L1 Qs 5 3 pfdz (12.20) 0 where p 5 shaft perimeter 5 pDs f 5 unit frictional (or skin) resistance 5 Ksor tan dr (12.21) K 5 earth pressure coefficient < Ko 5 1 2 sin fr (12.22) sor 5 effective vertical stress at any depth z Thus, L1 L1 Qs 5 3 pfdz 5 pDs (1 2 sin fr ) 3 sor tan dr dz 0 (12.23) 0 The value of sor will increase to a depth of about 15Ds and will remain constant thereafter, as shown in Figure 11.16. For cast-in-pile concrete and good construction techniques, a rough interface develops and, hence, dr>fr may be taken to be one. With poor slurry construction, dr>fr < 0.7 to 0.8. Allowable Net Load, Qall (net) An appropriate factor of safety should be applied to the ultimate load to obtain the net allowable load, or Qall(net) 5 12.8 Qp(net) 1 Qs FS (12.24) Load-Bearing Capacity Based on Settlement On the basis of a database of 41 loading tests, Reese and O’Neill (1989) proposed a method for calculating the load-bearing capacity of drilled shafts that is based on settlement. The method is applicable to the following ranges: 1. 2. 3. 4. Shaft diameter: Ds 5 0.52 to 1.2 m Bell depth: L 5 4.7 to 30.5 m Field standard penetration resistance: N60 5 5 to 60 Concrete slump 5 100 to 225 mm 12.8 Load-Bearing Capacity Based on Settlement 653 Qu 1.5 m (5 ft) noncontributing zone (cohesive soil only) i1 i2 Li Ds L fi p Li ii L1 Noncontributing zones: Length Ds (cohesive soil only) iN Db No side-load transfer permitted on perimeter of bell qp Ap Figure 12.11 Development of Eq. (12.25) Reese and O’Neill’s procedure (see Figure 12.11) gives N Qu(net) 5 a fipDLi 1 qpA p (12.25) i51 where fi 5 ultimate unit shearing resistance in layer i p 5 perimeter of the shaft 5 pDs qp 5 unit point resistance A p 5 area of the base 5 (p>4)D2b fi 5 b1sozi r , b2 (12.26) r 5 vertical effective stress at the middle of layer i. where sozi b1 5 b3 2 b4z0.5 i (for 0.25 # b1 # 1.2) (12.27) 654 Chapter 12: Drilled-Shaft Foundations The units for fi, zi, and sozi r and the magnitude of b2, b3, and b4 in the SI system are Item SI fi zi sozi r b2 b3 b4 kN>m2 m kN>m2 192 kN>m2 1.5 0.244 The point bearing capacity is qp 5 b5N60 # b6 3for Db , 1.27 m4 (12.28) where N60 5 field standard penetration number within a distance of 2Db below the base of the drilled shaft. The magnitudes of b5 and b6 and the unit of qp in the SI system are given here. Item SI b5 b6 qp 57.5 4310 kN>m2 kN>m2 If Db is equal to or greater than 1.27 m, excessive settlement may occur. In that case, qp may be replaced by qpr or qpr 5 1.27 q Db (m) p (12.29) Based on the desired level of settlement, Figures 12.12 and 12.13 may now be used to calculate the allowable load, Qall(net). Note that the trend lines given in these figures is the average of all test results. More recently, Rollins et al. (2005) have modified Eq. (12.27) for gravelly sands as follows: For sand with 25 to 50% gravel, (for 0.25 # b1 # 1.8) b1 5 b7 2 b8z0.75 i (12.30) 12.8 Load-Bearing Capacity Based on Settlement 655 2.0 End bearing Ultimate end bearing, qp Ap 1.6 1.2 Trend line 0.8 0.4 0 0 2 4 6 8 Settlement of base (%) Diameter of base, Db 10 12 Figure 12.12 Normalized base-load transfer versus settlement in sand Side-load transfer Ultimate side-load transfer, fi p Li 1.2 1.0 Trend line 0.8 0.6 0.4 0.2 0 0 0.4 0.8 1.2 1.6 Settlement (%) Diameter of shaft, Ds 2.0 Figure 12.13 Normalized side-load transfer versus settlement in sand For sand with more than 50% gravel, b1 5 b9e2b10zi (for 0.25 # b1 # 3.0) (12.31) The magnitudes of b7, b8, b9, and b10 and the unit of zi in the SI system are given here. 656 Chapter 12: Drilled-Shaft Foundations Item SI b7 b8 b9 b10 zi 2.0 0.15 3.4 2 0.085 m Figure 12.14 provides the normalized side-load transfer trend based on the desired level of settlement for gravelly sand and gravel. Side-load transfer Ultimate side-load transfer, fi p Li 1.0 0.8 0.6 Trend line 0.4 0.2 0 0 0.4 0.8 1.2 1.6 Settlement of base (%) Diameter of shaft, Ds (a) 2.0 2.4 2.0 2.4 Side-load transfer Ultimate side-load transfer, fi p Li 1.0 0.8 Trend line 0.6 0.4 0.2 0 0 0.4 0.8 1.2 1.6 Settlement of base (%) Diameter of shaft, Ds (b) Figure 12.14 Normalized side-load transfer versus settlement: (a) gravelly sand (gravel 25–50%) and (b) gravel (more than 50%) 12.8 Load-Bearing Capacity Based on Settlement 657 Example 12.1 A soil profile is shown in Figure 12.15. A point bearing drilled shaft with a bell is placed in a layer of dense sand and gravel. Determine the allowable load the drilled shaft could carry. Use Eq. (12.5) and a factor of safety of 4. Take Ds 5 1 m and Db 5 1.75 m. For the dense sand layer, fr 5 36°; Es 5 500pa . Ignore the frictional resistance of the shaft. Solution We have Qp(net) 5 A p 3qr(Nq 2 1)FqsFqdFqc4 and qr 5 (6) (16.2) 1 (2) (19.2) 5 135.6 kN>m2 For fr 5 36°, from Table 12.1, Nq 5 37.75. Also, Fqs 5 1.727 and Fqd 5 1 1 C tan21 a L ≤ Db 5 1 1 0.247 tan21 ¢ 8 ≤ 5 1.335 1.75 Qu 6m Ds Loose sand ␥ 16.2 kN/m3 Dense sand and gravel ␥ 19.2 kN/m3 36˚ 2m Db Figure 12.15 Allowable load of drilled shaft 658 Chapter 12: Drilled-Shaft Foundations From Eq. (12.9), Icr 5 0.5 expB2.85 cot¢ 45 2 fr ≤ R 5 134.3 (See Table 12.1) 2 From Eq. (12.12), Es 5 mpa . With m 5 500, we have Es 5 (500) (100) 5 50,000 kN>m2 From Eq. (12.13) and Table 12.1, ms 5 0.265 So Ir 5 Es 50,000 5 5 200.6 2(1 1 ms ) (qr) (tan fr) 2(1 1 0.265) (135.6) (tan 36) From Eq. (12.10), Irr 5 Ir 1 1 Ir D with D5n qr 135.6 5 0.00225¢ ≤ 5 0.0031 pa 100 it follows that Irr 5 200.6 5 123.7 1 1 (200.6) (0.0031) Irr is less than Icr . So, from Eq. (12.17), Fqc 5 exp b (23.8 tan fr) 1 B (3.07 sin fr) (log 10 2Irr ) Rr 1 1 sin fr 5 exp b (23.8 tan 36) 1 B (3.07 sin 36) log(2 3 123.7) R r 5 0.958 1 1 sin 36 Hence, p Qp(net) 5 B¢ ≤ (1.75) 2R (135.6) (37.75 2 1) (1.727) (1.335) (0.958) 5 26,474 kN 4 and Q p(all) 5 Qp(net) FS 5 26,474 < 6619 kN 4 ■ 12.8 Load-Bearing Capacity Based on Settlement 659 Example 12.2 Solve Example 12.1 using Eq. (12.18). Solution Equation (12.18) asserts that Qp(net) 5 A pqr(vN q* 2 1) We have (also see Table 12.2) N q* 5 0.21e0.17fr 5 0.21e (0.17)(36) 5 95.52 and L 8 5 5 4.57 Db 1.75 From Figure 12.10, for fr 5 36° and L>Db 5 4.57, the value of v is about 0.83. So p Qp(net) 5 B ¢ ≤ (1.75) 2 R (135.6) 3(0.83) (95.52) 2 14 5 25,532 kN 4 and Qp(all) 5 25,532 5 6383 kN 4 ■ Example 12.3 A drilled shaft is shown in Figure 12.16. The uncorrected average standard penetration number (N60 ) within a distance of 2Db below the base of the shaft is about 30. Determine a. The ultimate load-carrying capacity b. The load-carrying capacity for a settlement of 12mm. Use Eq. (12.30). Solution Part a From Eqs. (12.26) and (12.27), fi 5 b1sozi r and b1 5 2.0 2 0.15z0.75 For this problem, zi 5 6>2 5 3 m, so b 5 2 2 (0.15) (3) 0.75 5 1.658 660 Chapter 12: Drilled-Shaft Foundations Loose sandy gravel ␥ 16 kN/m3 6m 1m 1m 1.5 m Dense sandy gravel ␥ 19 kN/m3 N60 30 Figure 12.16 Drilled shaft supported by a dense layer of sandy gravel and sozi r 5 gzi 5 (16) (3) 5 48 kN>m2 Thus, fi 5 (48) (1.658) 5 79.58 kN>m2 and Sfi p DLi 5 (79.58) (p 3 1) (6) 5 1500 kN From Eq. (12.28), qp 5 57.5N60 5 (57.5) (30) 5 1725 kN>m2 Note that Db is greater than 1.27. So we will use Eq. (12.29a). qpr 5 a 1.27 1.27 bqp 5 ¢ ≤ (1725) < 1461 kN>m2 Db 1.5 Now qprA p 5 (14.61) ¢ p 3 1.52 ≤ < 2582 kN 4 Hence, Qult(net) 5 qprA p 1 Sfi p DLi 5 2582 1 1500 5 4082 kN 12.9 Drilled Shafts in Clay: Load-Bearing Capacity 661 Part b We have Allowable settlement 12 5 5 0.12 5 1.2% Ds (1.0) (1000) The trend line in Figure 12.14a shows that, for a normalized settlement of 1.2%, the normalized load is about 0.8 . Thus, the side-load transfer is (0.8) (1500) < 1200 kN. Similarly, Allowable settlement 12 5 5 0.008 5 0.8% Db (1.5) (1000) The trend line shown in Figure 12.12 indicates that, for a normalized settlement of 1.4%, the normalized base load is 0.317. So the base load is (0.317) (2582) 5 818.5 kN. Hence, the total load is Q 5 1200 1 818.5 < 2018.5 kN 12.9 ■ Drilled Shafts in Clay: Load-Bearing Capacity For saturated clays with f 5 0, the bearing capacity factor Nq in Eq. (12.4) is equal to unity. Thus, for this case, Qp(net) < A pcuNcFcsFcdFcc (12.32) where cu 5 undrained cohesion. Assuming that L > 3Db , we can rewrite Eq. (12.32) as Qp(net) 5 A pcuN *c (12.33) where N c* 5 NcFcsFcdFcc 5 1.333(ln Ir ) 1 14 in which Ir 5 soil rigidity index. (12.34) The soil rigidity index was defined in Eq. (12.11). For f 5 0, Ir 5 Es 3cu (12.35) O’Neill and Reese (1999) provided an approximate relationship between cu and Es>3cu . This relationship is shown in Figure 12.17. For all practical purposes, if cu>pa is 662 Chapter 12: Drilled-Shaft Foundations 300 250 200 Based on O'Neill and Reese (1999) Es 150 3cu 100 50 0 0 0.5 1.0 cu pa 1.5 2.0 Figure 12.17 Approximate Es variation of with cu>pa 3cu (Note: pa 5 atmospheric pressure) : (Based on O’Neill and Reese, 1999) equal to or greater than unity (pa 5 atmospheric pressure < 100 kN>m2), then the magnitude of N c* can be taken to be 9. Experiments by Whitaker and Cooke (1966) showed that, for belled shafts, the full value of N c* 5 9 is realized with a base movement of about 10 to 15% of Db . Similarly, for straight shafts (Db 5 Ds ), the full value of N c* 5 9 is obtained with a base movement of about 20% of Db . The expression for the skin resistance of drilled shafts in clay is similar to Eq. (11.55), or L5L1 Qs 5 a a*cup DL (12.36) L50 Kulhawy and Jackson (1989) reported the field-test result of 106 straight drilled shafts—65 in uplift and 41 in compression. The best correlation obtained from the results is a* 5 0.21 1 0.25¢ pa ≤ <1 cu (12.37) where pa 5 atmospheric pressure < 100 kN>m2. So, conservatively, we may assume that a* 5 0.4 (12.38) 12.10 Load-Bearing Capacity Based on Settlement 12.10 663 Load-Bearing Capacity Based on Settlement Reese and O’Neill (1989) suggested a procedure for estimating the ultimate and allowable (based on settlement) bearing capacities for drilled shafts in clay. According to this procedure, we can use Eq. (12.25) for the net ultimate load, or n Qult(net) 5 a fip DLi 1 qpA p i51 The unit skin friction resistance can be given as fi 5 ai*cu(i) (12.39) The following values are recommended for a*i : a*i 5 0 for the top 1.5 m (5 ft) and bottom 1 diameter, Ds, of the drilled shaft. (Note: If Db . Ds , then a* 5 0 for 1 diameter above the top of the bell and for the peripheral area of the bell itself.) a*i 5 0.55 elsewhere. The expression for qp (point load per unit area) can be given as qp 5 6cub a1 1 0.2 L b # 9cub # 40pa Db (12.40) where cub 5 average undrained cohesion within a vertical distance of 2Db below the base pa 5 atmospheric pressure If Db is large, excessive settlement will occur at the ultimate load per unit area, qp, as given by Eq. (12.40). Thus, for Db . 1.91 m, qp may be replaced by qpr 5 Frqp (12.41) 2.5 #1 c1Db 1 c2 (12.42) where Fr 5 The relations for c1 and c2 along with the unit of Db in the SI system are given in Table 12.3. Figures 12.18 and 12.19 may now be used to evaluate the allowable load-bearing capacity, based on settlement. (Note that the ultimate bearing capacity in Figure 12.18 is qp , not qpr .) To do so, 664 Chapter 12: Drilled-Shaft Foundations Step 1. Select a value of settlement, s. N Step 2. Calculate a fip DLi and qpA p . i51 Step 3. Using Figures 12.18 and 12.19 and the calculated values in Step 2, determine the side load and the end bearing load. Step 4. The sum of the side load and the end bearing load gives the total allowable load. Table 12.3 Relationships for c1 and c2 Item SI c1 5 2.78 3 1024 1 8.26 3 1025 a c1 c2 c2 5 0.0653cub (kN>m2 )4 0.5 (0.5 # c2 # 1.5) Db mm L b # 5.9 3 1024 Db 1.2 Side-load transfer Ultimate side-load transfer, fi p Li 1.0 0.8 Trend line 0.6 0.4 0.2 0 0 0.4 0.8 1.2 1.6 Settlement (%) Diameter of shaft, Ds 2.0 Figure 12.18 Normalized side-load transfer versus settlement in cohesive soil 12.10 Load-Bearing Capacity Based on Settlement 665 1.0 End bearing Ultimate end bearing, qp Ap 0.8 0.6 Trend line 0.4 0.2 0 0 2 4 6 Settlement of base (%) Diameter of base, Db 8 10 Figure 12.19 Normalized base-load transfer versus settlement in cohesive soil Example 12.4 Figure 12.20 shows a drilled shaft without a bell. Here, L1 5 8.23 m, L2 5 2.59 m, Ds 5 1.0 m, cu(1) 5 50 kN>m2, and cu(2) 5 108.75 kN>m2. Determine a. The net ultimate point bearing capacity b. The ultimate skin resistance c. The working load, Qw (FS 5 3) Use Eqs. (12.33), (12.36), and (12.38). Solution Part a From Eq. (12.33), p Qp(net) 5 A pcuN c* 5 A pcu(2)N c* 5 B ¢ ≤ (1) 2 R (108.75) (9) 5 768.7 kN 4 (Note: Since cu(2)>pa . 1, N c* < 9.) 666 Chapter 12: Drilled-Shaft Foundations Clay L1 cu (1) Ds Clay L2 cu (2) Figure 12.20 A drill shaft without a bell Part b From Eq. (12.36), Qs 5 Sa*cupDL From Eq. (12.38), a* 5 0.4 p 5 pDs 5 (3.14) (1.0) 5 3.14 m and Qs 5 (0.4) (3.14) 3(50 3 8.23) 1 (108.75 3 2.59) 4 5 871 kN Part c Qw 5 Qp(net) 1 Qs FS 5 768.7 1 871 5 546.6 kN 3 ■ Example 12.5 A drilled shaft in a cohesive soil is shown in Figure 12.21. Use Reese and O’Neill’s method to determine the following. a. The ultimate load-carrying capacity. b. The load-carrying capacity for an allowable settlement of 12 mm. 12.10 Load-Bearing Capacity Based on Settlement Clay cu(1) 40 kN/m2 0.76 m 3m 6m 3m Clay cu(2) 60 kN/m2 1.5 m Clay cu 145 kN/m2 1.2 m Figure 12.21 A drilled shaft in layered clay Solution Part a From Eq. (12.39), fi 5 ai*cu(i) From Figure 12.21, DL1 5 3 2 1.5 5 1.5 m DL2 5 (6 2 3) 2 Ds 5 (6 2 3) 2 0.76 5 2.24 m cu(1) 5 40 kN>m2 and cu(2) 5 60 kN>m2 Hence, SfipDLi 5 Sai*cu(i)pDLi 5 (0.55) (40) (p 3 0.76) (1.5) 1 (0.55) (60) (p 3 0.76) (2.24) 5 255.28 kN 667 668 Chapter 12: Drilled-Shaft Foundations Again, from Eq. (12.40), qp 5 6cub ¢1 1 0.2 L 6 1 1.5 ≤ 5 (6) (145) B1 1 0.2¢ ≤ R 5 1957.5 kN>m2 Db 1.2 A check reveals that qp 5 9cub 5 (9) (145) 5 1305 kN>m2 , 1957.5 kN>m2 So we use qp 5 1305 kN>m2 p p qpA p 5 qp ¢ D2b ≤ 5 (1305) B ¢ ≤ (1.2) 2 R 5 1475.9 kN 4 4 Hence, Qult 5 Sai*cu(i)pDLi 1 qpA p 5 255.28 1 1475.9 < 1731 kN Part b We have 12 Allowable settlement 5 5 0.0158 5 1.58% Ds (0.76) (1000) The trend line shown in Figure 12.18 indicates that, for a normalized settlement of 1.58%, the normalized side load is about 0.9. Thus, the side load is (0.9) (SfipDLi ) 5 (0.9) (255.28) 5 229.8 kN Again, Allowable settlement 12 5 5 0.01 5 1.0% Db (1.2) (1000) The trend line shown in Figure 12.19 indicates that, for a normalized settlement of 1.0%, the normalized end bearing is about 0.63, so Base load 5 (0.63) (qpA p ) 5 (0.63) (1475.9) 5 929.8 kN Thus, the total load is Q 5 229.8 1 929.8 5 1159.6 kN 12.11 ■ Settlement of Drilled Shafts at Working Load The settlement of drilled shafts at working load is calculated in a manner similar to that outlined in Section 11.15. In many cases, the load carried by shaft resistance is small compared with the load carried at the base. In such cases, the contribution of s3 may be ignored. (Note that in Eqs. (11.74) and (11.75) the term D should be replaced by Db for drilled shafts.) 12.11 Settlement of Drilled Shafts at Working Load 669 Example 12.6 Refer to Figure 12.20. Given: L1 5 8 m, L2 5 3 m, Ds 5 1.5 m, cu(1) 5 50 kN>m2, cu(2) 5 105 kN>m2, and working load Qw 5 1005 kN. Estimate the elastic settlement at the working load. Use Eqs. (11.73), (11.75), and (11.76). Take j 5 0.65, Ep 5 21 3 106 kN>m2, Es 5 14,000 kN>m2, ms 5 0.3, and Qwp 5 250 kN. Solution From Eq. (11.73), se(1) 5 (Qwp 1 jQ ws )L A pEp Now, Qws 5 1005 2 250 5 755 kN so se(1) 5 3250 1 (0.65) (755)4 (11) p ¢ 3 1.52 ≤ (21 3 106 ) 4 5 0.00022 m 5 0.22 mm From Eq. (11.75), se(2) 5 QwpCp Dbqp From Table 11.13, for stiff clay, Cp < 0.04; also, qp 5 cu(b)N c* 5 (105) (9) 5 945 kN>m2 Hence, se(2) 5 (250) (0.04) 5 0.0071 m 5 7.1 mm (1.5) (945) Again, from Eqs. (11.76) and (11.77), se(3) 5 ¢ Qws Ds ≤ ¢ ≤ (1 2 m2s )Iws pL Es where L 11 5 2 1 0.35 5 2.95 Å Ds Å 1.5 Iws 5 2 1 0.35 se(3) 5 B 755 1.5 R¢ ≤ (1 2 0.32 ) (2.95) 5 0.0042 m 5 4.2 mm (p 3 1.5) (11) 14,000 670 Chapter 12: Drilled-Shaft Foundations The total settlement is se 5 se(1) 1 se(2) 1 se(3) 5 0.22 1 7.1 1 4.2 < 11.52 mm 12.12 ■ Lateral Load-Carrying Capacity—Characteristic Load and Moment Method Several methods for analyzing the lateral load-carrying capacity of piles, as well as the load-carrying capacity of drilled shafts, were presented in Section 11.16; therefore, they will not be repeated here. In 1994, Duncan et al. developed a characteristic load method for estimating the lateral load capacity for drilled shafts that is fairly simple to use. We describe this method next. According to the characteristic load method, the characteristic load Qc and moment Mc form the basis for the dimensionless relationship that can be given by the following correlations: Characteristic Load Qc 5 7.34D2s (EpRI ) ¢ Qc 5 1.57D2s (EpRI ) ¢ cu 0.68 ≤ EpRI grDsfrKp EpRI (for clay) (12.43) (for sand) (12.44) (for clay) (12.45) (for sand) (12.46) 0.57 ≤ Characteristic Moment Mc 5 3.86D3s (EpRI ) ¢ Mc 5 1.33D3s (EpRI ) ¢ cu 0.46 ≤ EpRI grDsfrKp EpRI 0.40 ≤ In these equations, Ds 5 diameter of drilled shafts Ep 5 modulus of elasticity of drilled shafts RI 5 ratio of moment of inertia of drilled shaft section to moment of inertia of a solid section (Note: RI 5 1 for uncracked shaft without central void) gr 5 effective unit weight of sand fr 5 effective soil friction angle (degrees) Kp 5 Rankine passive pressure coefficient 5 tan2 (45 1 fr>2) Deflection Due to Load Qg Applied at the Ground Line Figures 12.22 and 12.23 give the plot of Qg>Qc versus xo>Ds for drilled shafts in sand and clay due to the load Qg applied at the ground surface. Note that xo is the ground 12.12 Lateral Load-Carrying Capacity—Characteristic Load and Moment Method 0.050 0.050 0.045 0.045 671 Qg Fixed head Qc 0.030 0.030 Qg Free head Qc Mg Mc Qg Qc Mg Mc 0.015 xo Ds 0.015 Qg Qg Free Fixed Qc Qc 0.005 0.010 0.020 0.0065 0.0091 0.0135 0.0133 0.0197 0.0289 Mg Mc 0.0024 0.0048 0.0074 0 0 0 0.05 0.10 0.15 xo Ds Figure 12.22 Plot of Qg Qc and Mg Mc versus xo in clay Ds line deflection. If the magnitudes of Qg and Qc are known, the ratio Qg>Qc can be calculated. The figure can then be used to estimate the corresponding value of xo>Ds and, hence, xo . Deflection Due to Moment Applied at the Ground Line Figures 12.22 and 12.23 give the variation plot of Mg>Mc with xo>Ds for drilled shafts in sand and clay due to an applied moment Mg at the ground line. Again, xo is the ground line deflection. If the magnitudes of Mg , Mc , and Ds are known, the value of xo can be calculated with the use of the figure. 672 Chapter 12: Drilled-Shaft Foundations 0.015 0.015 Mg Mc Qg Fixed head Qc 0.010 0.010 Qg Free head Qc Qg Qc 0.005 Mg Mc xo Ds Qg Free Qc Qg Fixed Qc Mg Mc 0.005 0.010 0.020 0.0013 0.0021 0.0033 0.0028 0.0049 0.0079 0.0009 0.0019 0.0032 0.005 0 0 0 0.05 0.10 0.15 xo Ds Figure 12.23 Plot of Qg Qc and Mg Mc versus xo in sand Ds Deflection Due to Load Applied Above the Ground Line When a load Q is applied above the ground line, it induces both a load Qg 5 Q and a moment Mg 5 Qe at the ground line, as shown in Figure 12.24a. A superposition solution can now be used to obtain the ground line deflection. The step-by-step procedure is as follows (refer to Figure 12.24b): Step 1. Calculate Qg and Mg . Step 2. Calculate the deflection xoQ that would be caused by the load Qg acting alone. Step 3. Calculate the deflection xoM that would be caused by the moment acting alone. Step 4. Determine the value of a load QgM that would cause the same deflection as the moment (i.e., xoM). Step 5. Determine the value of a moment MgQ that would cause the same deflection as the load (i.e., xoQ). Step 6. Calculate (Qg 1 QgM )>Qc . and determine xoQM>Ds . Step 7. Calculate (Mg 1 MgQ )>Mc and determine xoMQ>Ds . 12.12 Lateral Load-Carrying Capacity—Characteristic Load and Moment Method Q 673 Mg Qe e Qg Q xo xoQM xoMQ 0.5 Ds Ds Ds ( Q ) (a) Qg Qc Mg Mc Qg QgM Qc QgM Qc Qg Qc Step 6 Mg Mc Step 4 Step 7 Step 4 Qg Qc Step 7 Mg MgQ Mc Step 2 Step 3 Step 3 Step 5 Mg Mc MgQ Mc Step 2 xoQ Ds xoM Ds xo Ds xoQM Ds xoMQ Ds (b) Figure 12.24 Superposition of deflection due to load and moment Step 8. Calculate the combined deflection: xo(combined) 5 0.5(xoQM 1 xoMQ ) (12.47) Maximum Moment in Drilled Shaft Due to Ground Line Load Only Figure 12.25 shows the plot of Qg>Qc with Mmax>Mc for fixed- and free-headed drilled shafts due only to the application of a ground line load Qg . For fixed-headed 674 Chapter 12: Drilled-Shaft Foundations 0.045 0.020 Fixed Free 0.015 Fixed Free 0.030 Qg Qc 0.010 Qg Qc (Sand) (Clay) 0.015 Clay 0.005 Sand 0 0 0.005 0 0.015 0.010 Mmax Mc Figure 12.23 Variation of Qg Qc with Mmax Mc shafts, the maximum moment in the shaft, Mmax , occurs at the ground line. For this condition, if Qc , Mc , and Qg are known, the magnitude of Mmax can be easily calculated. Maximum Moment Due to Load and Moment at Ground Line If a load Qg and a moment Mg are applied at the ground line, the maximum moment in the drilled shaft can be determined in the following manner: Step 1. Using the procedure described before, calculate xo(combined) from Eq. (12.47). Step 2. To solve for the characteristic length T, use the following equation: xo(combined) 5 2.43Qg EpIp T3 1 1.62Mg EpIp T2 (12.48) Step 3. The moment in the shaft at a depth z below the ground surface can be calculated as Mz 5 A mQgT 1 BmMg (12.49) 12.12 Lateral Load-Carrying Capacity—Characteristic Load and Moment Method 675 where A m , Bm 5 dimensionless moment coefficients (Matlock and Reese, 1961); see Figure 12.26. The value of the maximum moment Mmax can be obtained by calculating Mz at various depths in the upper part of the drilled shaft. The characteristic load method just described is valid only if L>Ds has a certain minimum value. If the actual L>Ds is less than (L>Ds ) min , then the ground line deflections will be underestimated and the moments will be overestimated. The values of (L>Ds ) min for drilled shafts in sand and clay are given in the following table: Clay Sand EpRI EpRI cu (L/Ds)min g9Dsf9Kp (L/Ds)min 1 3 105 3 3 105 1 3 106 3 3 106 6 10 14 18 1 3 104 4 3 104 2 3 105 8 11 14 Am , Bm 0 0.2 0.4 0.6 0.8 1.0 0 Am 0.5 Bm z 1.0 T 1.5 2.0 Figure 12.26 Variation of A m and Bm with z>T 68125_12_ch12_p637-685.qxd 3/12/10 11:38 AM Page 676 676 Chapter 12: Drilled-Shaft Foundations Example 12.7 A free-headed drilled shaft in clay is shown in Figure 12.27. Let Ep 5 22 3106 kN>m2. Determine a. b. c. d. The ground line deflection, xo(combined) The maximum bending moment in the drilled shaft The maximum tensile stress in the shaft The minimum penetration of the shaft needed for this analysis Solution We are given Ds 5 1 m cu 5 100 kN>m2 RI 5 1 Ep 5 22 3 106 kN>m2 and Ip 5 pD4s (p) (1) 4 5 5 0.049 m4 64 64 Part a From Eq. (12.43), Qc 5 7.34D2s (EpRI ) ¢ cu 0.68 ≤ EpRI 5 (7.34) (1) 2 3(22 3 106 ) (1)4 B 0.68 100 R (22 3 106 ) (1) 5 37,607 kN Mg 200 kN-m Qg 150 kN Clay cu 100 kN/m2 Ds 1m Figure 12.27 Free-headed drilled shaft 12.12 Lateral Load-Carrying Capacity—Characteristic Load and Moment Method 677 From Eq. (12.45), Mc 5 3.86D3s (EpRI ) ¢ cu 0.46 ≤ EpRI 5 (3.86) (1) 3 3(22 3 106 ) (1)4 B 0.46 100 R 6 (22 3 10 ) (1) 5 296,139 kN-m Thus, Qg 5 Qc 150 5 0.004 37,607 From Figure 12.22, xoQ < (0.0025)Ds 5 0.0025 m 5 2.5 mm. Also, Mg Mc 5 200 5 0.000675 296,139 From Figure 12.22, xoM < (0.0014)Ds 5 0.0014 m 5 1.4 mm, so xoM 0.0014 5 5 0.0014 Ds 1 From Figure 12.22, for xoM>Ds 5 0.0014, the value of QgM>Qc < 0.002. Hence, xoQ Ds 5 0.0025 5 0.0025 1 From Figure 12.22, for xoQ>Ds 5 0.0025, the value of MgQ>Mc < 0.0013, so Qg Qc 1 QgM Qc 5 0.004 1 0.002 5 0.006 From Figure 12.22, for (Qg 1 QgM )>Qc 5 0.006, the value of xoQM>Ds < 0.0046. Hence, xoQM 5 (0.0046) (1) 5 0.0046 m 5 4.6 mm Thus, we have Mg Mc 1 MgQ Mc 5 0.000675 1 0.0013 < 0.00198 From Figure 12.22, for (Mg 1 MgQ )>Mc 5 0.00198, the value of xoMQ>Ds < 0.0041. Hence, xoMQ 5 (0.0041) (1) 5 0.0041 m 5 4.1 mm 678 Chapter 12: Drilled-Shaft Foundations Consequently, xo (combined) 5 0.5(xoQM 1 xoMQ ) 5 (0.5) (4.6 1 4.1) 5 4.35 mm Part b From Eq. (12.48), 2.43Qg xo (combined) 5 EpIp T3 1 1.62Mg EpIp T2 so (2.43) (150) 0.00435 m 5 6 (22 3 10 ) (0.049) T3 1 (1.62) (200) (22 3 106 ) (0.049) T2 or 0.00435 m 5 338 3 1026 T3 1 300.6 3 1026 T2 and it follows that T < 2.05 m From Eq. (12.49), Mz 5 A m Qg T 1 Bm Mg 5 A m (150) (2.05) 1 Bm (200) 5 307.5A m 1 200 Bm Now the following table can be prepared: z T Am (Figure 12.26) 0 0.4 0.6 0.8 1.0 1.1 1.25 Bm (Figure 12.26) 0 0.36 0.52 0.63 0.75 0.765 0.75 Mz (kN-m) 1.0 0.98 0.95 0.9 0.845 0.8 0.73 200 306.7 349.9 373.7 399.6 395.2 376.6 So the maximum moment is 399.4 kN-m < 400 kN-m and occurs at z>T < 1. Hence, z 5 (1) (T) 5 (1) (2.05 m) 5 2.05 m Part c The maximum tensile stress is Mmax ¢ s tensile 5 Ip Ds ≤ 2 5 1 (400) ¢ ≤ 2 0.049 5 4081.6 kN , m2 12.13 Drilled Shafts Extending into Rock 679 Part d We have EpRI cu 5 (22 3 106 ) (1) 5 2.2 3 105 100 By interpolation, for (EpRI )>cu 5 2.2 3 105, the value of (L>D s ) min < 8.5. So L < (8.5) (1) 5 8.5 m 12.13 ■ Drilled Shafts Extending into Rock In Section 12.1, we noted that drilled shafts can be extended into rock. Figure 12.28 shows a drilled shaft whose depth of embedment in rock is equal to L. When considering drilled shafts in rock, we can find several correlations between the end bearing capacity and the unconfined compression strength of intact rocks, qu. It is important to recognize that, in the field, there are cracks, joints, and discontinuities in the rock, and the influence of those factors should be considered. Keeping this in mind, Zhang and Einstein (1998) analyzed a data base of 39 full-scale drilled shaft tests in which the Qu Soil Rock z f L f unit side resistance qp unit point bearing Ds Db qp Figure 12.28 Drilled shaft socketed into rock 680 Chapter 12: Drilled-Shaft Foundations shaft bases were cast on or in generally soft rock with some degree of jointing. Based on these results, they proposed Qu(net) 5 Qp 1 Qs 5 qpA p 1 fpL (12.50) where end bearing capacity Qp can be expressed as Qp (MN) 5 qpA p 5 34.83(qu MN>m2 ) 0.5143A p (m2 )4 (12.51) Figure 12.29 shows the plot of qp (MN>m2 ) versus qu (MN>m2 ) obtained from the data on which Eq. (12.51) is based. Also, the side resistance Qs is Qs (MN) 5 fpL 5 30.4(qu MN>m2 ) 0.543pDs (m)43L(m) 4 (for smooth socket) (12.52) Qs (MN) 5 fpL 5 30.8(qu MN>m2 ) 0.543pDs (m)43L(m) 4 (for rough socket) (12.53) and 100 qp(MN/m2) 30 10 qp = 4.83 (qu)0.51 3 1 0.1 30 0.3 1 3 10 Unconfined compression strength, qu (MN/m2) Figure 12.29 Plot of qp versus qu (Adapted from Zhang and Einstein, 1998) 100 Problems 681 Example 12.8 Figure 12.30 shows a drilled shaft extending into a shale formation. For the intact rock cores, given qu 5 4.2 MN>m2. Estimate the allowable load-bearing capacity of the drilled shaft. Use a factor of safety (FS) 5 3. Assume a smooth socket for side resistance. Solution From Eq. (12.51), Qp 5 A p 34.83(qu ) 0.514 5 p (1) 2 3(4.83) (4.2) 0.514 5 7.89 MN 4 Again, from Eq. (12.52), Qs 5 0.4(qu ) 0.5 (pDsL) 5 0.4(4.2) 0.5 3(p) (1) (4)4 5 10.3 MN Hence, Qall 5 3m Qp 1 Qs Qu 7.89 1 10.3 5 5 5 6.06MN FS FS 3 Soft clay Ds 1 m Shale Smooth socket 4m Drilled shaft Figure 12.30 Drilled shaft extending into rock ■ Problems 12.1 A drilled shaft is shown in Figure P12.1. Determine the net allowable point bearing capacity. Given gc 5 15.6 kN>m3 Db 5 2 m Ds 5 1.2 m gs 5 17.6 kN>m3 L1 5 6 m fr 5 35° L2 5 3 m cu 5 35 kN>m2 Factor of safety 5 3 Use Eq. (12.18). 12.2 Redo Problem 12.1, this time using Eq. (12.15). Let Es 5 600pa . 12.3 For the drilled shaft described in Problem 12.1, what skin resistance would develop in the top 6 m, which are in clay? Use Eqs. (12.36) and (12.38). 682 Chapter 12: Drilled-Shaft Foundations L1 Silty clay Ds ␥c cu Sand ␥s c 0 L2 Db Figure P12.1 12.4 Redo Problem 12.1 with the following: Db 5 1.75 m gc 5 17.8 kN>m3 Ds 5 1 m gs 5 18.2 kN>m3 L1 5 6.25 m fr 5 32° L2 5 2.5 m cu 5 32 kN>m2 Factor of safety 5 4 12.5 Redo Problem 12.4 using Eq. (12.5). Let Es 5 400pa . 12.6 For the drilled shaft described in Problem 12.4, what skin friction would develop in the top 6.25 m? a. Use Eqs. (12.36) and (12.38). b. Use Eq. (12.39). 12.7 Figure P12.7 shows a drilled shaft without a bell. Assume the following values: cu(1) 5 50 kN>m2 L1 5 6 m L2 5 7 m cu(2) 5 75 kN>m2 Ds 5 1.5 m Determine: a. The net ultimate point bearing capacity [use Eqs. (12.33) and (12.34)] b. The ultimate skin friction [use Eqs. (12.36) and (12.38)] c. The working load Qw (factor of safety 5 3) 12.8 Repeat Problem 12.7 with the following data: L1 5 6.1 m cu(1) 5 70 kN>m2 L2 5 3.05 m cu(2) 5 120 kN>m2 Ds 5 0.91 m Use Eqs. (12.39) and (12.40). 12.9 A drilled shaft in a medium sand is shown in Figure P12.9. Using the method proposed by Reese and O’Neill, determine the following: Problems L1 683 Clay Ds cu(1) Clay L2 cu(2) Figure P12.7 a. The net allowable point resistance for a base movement of 25 mm b. The shaft frictional resistance for a base movement of 25 mm c. The total load that can be carried by the drilled shaft for a total base movement of 25 mm Assume the following values: g 5 18 kN>m3 fr 5 38° Dr 5 65%(medium sand) L 5 12 m L1 5 11 m Ds 5 1 m Db 5 2 m Ds Medium sand L1 L ␥ Average standard penetration number (N60) within 2Db below the drilled shaft 19 Db Figure P12.9 684 Chapter 12: Drilled-Shaft Foundations 12.10 In Figure P12.9, let L 5 7 m, L1 5 6 m, Ds 5 0.75 m, Db 5 1.25 m, g 5 18 kN>m3, and fr 5 37°. The average uncorrected standard penetration number (N60 ) within 2Db below the drilled shaft is 29. Determine a. The ultimate load-carrying capacity b. The load-carrying capacity for a settlement of 12 mm. The sand has 35% gravel. Use Eq. (12.30) and Figures 12.12 and 12.14. 12.11 For the drilled shaft described in Problem 12.7, determine a. The ultimate load-carrying capacity b. The load carrying capacity for a settlement of 25 mm Use the procedure outlined by Reese and O’Neill. (See Figures 12.18 and 12.19.) 12.12 For the drilled shaft described in Problem 12.7, estimate the total elastic settlement at working load. Use Eqs. (11.73), (11.75), and (11.76). Assume that Ep 5 20 3 106 kN>m2, Cp 5 0.03, j 5 0.65, ms 5 0.3, Es 5 12,000 kN>m2, and Q ws 5 0.8Qw . Use the value of Qw from Part (c) of Problem 12.7. 12.13 For the drilled shaft described in Problem 12.8, estimate the total elastic settlement at working load. Use Eqs. (11.73), (11.75), and (11.76). Assume that Ep 5 3 3 106 lb>in2, Cp 5 0.03, j 5 0.65, ms 5 0.3, Es 5 2000 lb>in2, and Q ws 5 0.83Qw . Use the value of Qw from Part (c) of Problem 12.8. 12.14 Figure P12.14 shows a drilled shaft extending into clay shale. Given: qu (clay shale) 5 1.81 MN>m2. Considering the socket to be rough, estimate the allowable load-carrying capacity of the drilled shaft. Use FS 5 4. 12.15 A free-headed drilled shaft is shown in Figure P12.15. Let Qg 5 260 kN, Mg 5 0, g 5 17.5 kN>m3, fr 5 35°, cr 5 0, and Ep 5 22 3 106 kN>m2. Determine a. The ground line deflection, xo b. The maximum bending moment in the drilled shaft c. The maximum tensile stress in the shaft d. The minimum penetration of the shaft needed for this analysis Mg 2m 1.5 m Loose sand ␥ 15 kN/m3 30˚ Qg Clay shale 8m Figure P12.14 Concrete drilled shaft Ds 1.25 m Figure P12.15 ␥ c, cu , References 685 References BEREZANTZEV, V. G., KHRISTOFOROV, V. S., and GOLUBKOV, V. N. (1961). “Load Bearing Capacity and Deformation of Piled Foundations,” Proceedings, Fifth International Conference on Soil Mechanics and Foundation Engineering, Paris, Vol. 2, pp. 11–15. CHEN, Y.-J., and KULHAWY, F. H. (1994). “Case History Evaluation of the Behavior of Drilled Shafts under Axial and Lateral Loading,” Final Report, Project 1493-04, EPRI TR-104601, Geotechnical Group, Cornell University, Ithaca, NY, December. DUNCAN, J. M., EVANS, L. T., JR., and Ooi, P. S. K. (1994). “Lateral Load Analysis of Single Piles and Drilled Shafts,” Journal of Geotechnical Engineering, ASCE, Vol. 120, No. 6, pp. 1018–1033. KULHAWY, F. H., and JACKSON, C. S. (1989). “Some Observations on Undrained Side Resistance of Drilled Shafts,” Proceedings, Foundation Engineering: Current Principles and Practices, American Society of Civil Engineers, Vol. 2, pp. 1011–1025. MATLOCK, H., and REESE, L.C. (1961). “Foundation Analysis of Offshore Pile-Supported Structures,” in Proceedings, Fifth International Conference on Soil Mechanics and Foundation Engineering, Vol. 2, Paris, pp. 91–97. O’NEILL, M. W. (1997). Personal communication. O’NEILL, M.W., and REESE, L.C. (1999). Drilled Shafts: Construction Procedure and Design Methods, FHWA Report No. IF-99-025. REESE, L. C., and O’NEILL, M. W. (1988). Drilled Shafts: Construction and Design, FHWA, Publication No. HI-88-042. REESE, L. C., and O’NEILL, M. W. (1989). “New Design Method for Drilled Shafts from Common Soil and Rock Tests,” Proceedings, Foundation Engineering: Current Principles and Practices, American Society of Civil Engineers, Vol. 2, pp. 1026–1039. REESE, L. C., TOUMA, F. T., and O’NEILL, M. W. (1976). “Behavior of Drilled Piers under Axial Loading,” Journal of Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 102, No. GT5, pp. 493–510. ROLLINS, K. M., CLAYTON, R. J., MIKESELL, R. C., and BLAISE, B. C. (2005). “Drilled Shaft Side Friction in Gravelly Soils,” Journal of Geotechnical and Geoenvironmental Engineering, American Society of Civil Engineers, Vol. 131, No. 8, pp. 987–1003. WHITAKER, T., and COOKE, R. W. (1966). “An Investigation of the Shaft and Base Resistance of Large Bored Piles in London Clay,” Proceedings, Conference on Large Bored Piles, Institute of Civil Engineers, London, pp. 7–49. ZHANG, L., and EINSTEIN, H. H. (1998). “End Bearing Capacity of Drilled Shafts in Rock,” Journal of Geotechnical and Geoenvironmental Engineering, American Society of Civil Engineers, Vol. 124, No. 7, pp. 574–584. 68125_13_ch13_p686-721.qxd 13 13.1 3/12/10 11:43 AM Page 686 Foundations on Difficult Soils Introduction In many areas of the United States and other parts of the world, certain soils make the construction of foundations extremely difficult. For example, expansive or collapsible soils may cause high differential movements in structures through excessive heave or settlement. Similar problems can also arise when foundations are constructed over sanitary landfills. Foundation engineers must be able to identify difficult soils when they are encountered in the field. Although not all the problems caused by all soils can be solved, preventive measures can be taken to reduce the possibility of damage to structures built on them. This chapter outlines the fundamental properties of three major soil conditions—collapsible soils, expansive soils, and sanitary landfills—and methods of careful construction of foundations. Collapsible Soil 13.2 Definition and Types of Collapsible Soil Collapsible soils, which are sometimes referred to as metastable soils, are unsaturated soils that undergo a large change in volume upon saturation. The change may or may not be the result of the application of additional load. The behavior of collapsing soils under load is best explained by the typical void ratio effective pressure plot (e against log sr ) for a collapsing soil, as shown in Figure 13.1. Branch ab is determined from the consolidation test on a specimen at its natural moisture content. At an effective pressure level of swr , the equilibrium void ratio is e1 . However, if water is introduced into the specimen for saturation, the soil structure will collapse. After saturation, the equilibrium void ratio at the same effective pressure level swr is e2 ; cd is the branch of the e–log sr curve under additional load after saturation. Foundations that are constructed on such soils may undergo large and sudden settlement if the soil under them becomes saturated with an unanticipated supply of moisture. The moisture may come from any of several sources, such as (a) broken water pipelines, (b) leaky sewers, (c) drainage 686 13.3 Physical Parameters for Identification e1 Void ratio, e e2 Effective pressure, (log scale) w a b 687 Water added c d Figure 13.1 Nature of variation of void ratio with pressure for a collapsing soil from reservoirs and swimming pools, (d) a slow increase in groundwater, and so on. This type of settlement generally causes considerable structural damage. Hence, identifying collapsing soils during field exploration is crucial. The majority of naturally occurring collapsing soils are aeolian—that is, winddeposited sands or silts, such as loess, aeolic beaches, and volcanic dust deposits. The deposits have high void ratios and low unit weights and are cohesionless or only slightly cohesive. Loess deposits have silt-sized particles. The cohesion in loess may be the result of clay coatings surrounding the silt-size particles. The coatings hold the particles in a rather stable condition in an unsaturated state. The cohesion also may be caused by the presence of chemical precipitates leached by rainwater. When the soil becomes saturated, the clay binders lose their strength and undergo a structural collapse. In the United States, large parts of the Midwest and the arid West have such types of deposit. Loess deposits are also found over 15 to 20% of Europe and over large parts of China. The thickness of loess deposits can range up to about 10 m in the central United States. In parts of China it can be up to 100 m. Figure 13.2 shows the extent of the loess deposits in the Mississippi River basin. Many collapsing soils may be residual soils that are products of the weathering of parent rocks. Weathering produces soils with a large range of particle-size distribution. Soluble and colloidal materials are leached out by weathering, resulting in large void ratios and thus unstable structures. Many parts of South Africa and Zimbabwe have residual soils that are decomposed granites. Sometimes collapsing soil deposits may be left by flash floods and mudflows. These deposits dry out and are poorly consolidated. An excellent review of collapsing soils is that of Clemence and Finbarr (1981). 13.3 Physical Parameters for Identification Several investigators have proposed various methods for evaluating the physical parameters of collapsing soils for identification. Some of these methods are discussed briefly in Table 13.1. Jennings and Knight (1975) suggested a procedure for describing the collapse potential of a soil: An undisturbed soil specimen is taken at its natural moisture content in a consolidation ring. Step loads are applied to the specimen up to a pressure level swr of 200 kN>m2. (In Figure 13.1, this is swr .) At that pressure, the specimen 688 Chapter 13: Foundations on Difficult Soils Minnesota Wisconsin South Dakota Iowa Indiana Nebraska Illinois Kansas Missouri Kentucky Tennessee Arkansas Alabama Mississippi Louisiana Figure 13.2 Loess deposit in Mississippi River basin is flooded for saturation and left for 24 hours. This test provides the void ratios e1 and e2 before and after flooding, respectively. The collapse potential may now be calculated as Cp 5 De 5 e1 2 e2 1 1 eo (13.1) where eo 5 natural void ratio of the soil De 5 vertical strain The severity of foundation problems associated with a collapsible soil have been correlated with the collapse potential Cp by Jennings and Knight (1975). They were summarized by Clemence and Finbarr (1981) and are given in Table 13.2. Holtz and Hilf (1961) suggested that a loessial soil that has a void ratio large enough to allow its moisture content to exceed its liquid limit upon saturation is susceptible to collapse. So, for collapse, w(saturated) > LL (13.2) 13.3 Physical Parameters for Identification 689 where LL 5 liquid limit. However, for saturated soils, eo 5 wGs (13.3) where Gs 5 specific gravity of soil solids. Table 13.1 Reported Criteria for Identification of Collapsing Soila a Investigator Year Criteria Denisov 1951 Clevenger 1958 Priklonski 1952 Gibbs 1961 Soviet Building Code 1962 Feda 1964 Benites 1968 Handy 1973 Coefficient of subsidence: void ratio at liquid limit K5 natural void ratio K 5 0.5– 0.75: highly collapsible K 5 1.0: noncollapsible loam K 5 1.5 –2.0: noncollapsible soils If dry unit weight is less than 12.6 kN>m3, settlement will be large; if dry unit weight is greater than 14 kN>m3 settlement will be small. natural moisture content 2 plastic limit KD 5 plasticity index KD , 0: highly collapsible soils KD . 0.5: noncollapsible soils KD . 1.0: swelling soils saturation moisture content Collapse ratio, R 5 liquid limit This was put into graph form. eo 2 eL L5 1 1 eo where eo 5 natural void ratio and eL 5 void ratio at liquid limit. For natural degree of saturation less than 60%, if L . 20.1, the soil is a collapsing soil. wo PL 2 KL 5 Sr PI where wo 5 natural water content, Sr 5 natural degree of saturation, PL 5 plastic limit, and PI 5 plasticity index. For Sr , 100%, if KL . 0.85, the soil is a subsident soil. A dispersion test in which 2 g of soil are dropped into 12 ml of distilled water and specimen is timed until dispersed; dispersion times of 20 to 30 s were obtained for collapsing Arizona soils. Iowa loess with clay (,0.002 mm) contents: ,16%: high probability of collapse 16–24%: probability of collapse 24–32%: less than 50% probability of collapse .32%: usually safe from collapse Modified after Lutenegger and Saber (1988) 690 Chapter 13: Foundations on Difficult Soils Table 13.2 Relation of Collapse Potential to the Severity of Foundation Problemsa Cp (%) Severity of problem 0–1 1–5 5–10 10–20 20 No problem Moderate trouble Trouble Severe trouble Very severe trouble a From Clemence, S. P., and Finbarr, A. O. (1981). “Design Considerations for Collapsible Soils,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 107, GT3 pp. 305–317. With permission from ASCE. Combining Eqs. (13.2) and (13.3) for collapsing soils yields eo > (LL) (Gs ) (13.4) The natural dry unit weight of the soil required for its collapse is gd < Gsgw Gsgw 5 1 1 eo 1 1 (LL) (Gs ) (13.5) For an average value of Gs 5 2.65, the limiting values of gd for various liquid limits may now be calculated from Eq. (13.5). Figure 13.3 shows a plot of the preceding limiting values of dry unit weights against the corresponding liquid limits. For any soil, if the natural dry unit weight falls below the limiting line, the soil is likely to collapse. 20 Natural dry unit weight, ␥d (kN/m3) 18 16 14 Loessial soil likely to collapse 12 10 10 20 30 Liquid limit 40 45 Figure 13.3 Loessial soil likely to collapse 13.4 Procedure for Calculating Collapse Settlement 691 Care should be taken to obtain undisturbed samples for determining the collapse potentials and dry unit weights—preferably block samples cut by hand. The reason is that samples obtained by thin-walled tubes may undergo some compression during the sampling process. However, if cut block samples are used, the boreholes should be made without water. 13.4 Procedure for Calculating Collapse Settlement Jennings and Knight (1975) proposed the following laboratory procedure for determining the collapse settlement of structures upon saturation of soil: Step 1. Obtain two undisturbed soil specimens for tests in a standard consolidation test apparatus (oedometer). Step 2. Place the two specimens under 1 kN>m2 pressure for 24 hours. Step 3. After 24 hours, saturate one specimen by flooding. Keep the other specimen at its natural moisture content. Step 4. After 24 hours of flooding, resume the consolidation test on both specimens by doubling the load (the same as in the standard consolidation test) to the desired pressure level. Step 5. Plot the e–log sr graphs for both specimens (Figures 13.4a and b). Step 6. Calculate the in situ effective pressure, sor . Draw a vertical line corresponding to the pressure sor . Step 7. From the e–log sor curve of the soaked specimen, determine the preconsolidation pressure, scr . If scr >sor 5 0.8–1.5, the soil is normally consolidated; however, if scr >sor . 1.5, the soil is preconsolidated. o c o o , eo Specimen at natural moisture content e1 c o o , eo e1 e2 Soaked specimen Void ratio, e o log (a) log Specimen at natural moisture content e2 Void ratio, e Soaked specimen (b) Figure 13.4 Settlement calculation from double oedometer test: (a) normally consolidated soil; (b) overconsolidated soil 692 Chapter 13: Foundations on Difficult Soils Step 8. Determine eor , corresponding to sor from the e–log sor curve of the soaked specimen. (This procedure for normally consolidated and overconsolidated soils is shown in Figures 13.4a and b, respectively.) Step 9. Through point (sor , eor ) draw a curve that is similar to the e–log sor curve obtained from the specimen tested at its natural moisture content. Step 10. Determine the incremental pressure, Dsr, on the soil caused by the construction of the foundation. Draw a vertical line corresponding to the pressure of sor 1 Dsr in the e–log sr curve. Step 11. Now, determine De1 and De2 . The settlement of soil without change in the natural moisture content is Sc(1) 5 De1 (H) 1 1 eor (13.6) where H 5 thickness of soil susceptible to collapse. Also, the settlement caused by collapse in the soil structure is Sc(2) 5 (13.7) Foundation Design in Soils Not Susceptible to Wetting For actual foundation design purposes, some standard field load tests may also be conducted. Figure 13.5 shows the relation of the nature of load per unit arca versus settlement in a field load test in a loessial deposit. Note that the load–settlement relationship is essentially linear up to a certain critical pressure, scrr , at which there is a breakdown of the soil structure and hence a large settlement. Sudden breakdown of soil structure is more common with soils having a high natural moisture content than with normally dry soils. If enough precautions are taken in the field to prevent moisture from increasing under structures, spread foundations and mat foundations may be built on collapsible Load per unit area cr Settlement 13.5 De2 (H) 1 1 eor Figure 13.5 Field load test in loessial soil: load per unit area versus settlement 13.5 Foundation Design in Soils Not Susceptible to Wetting 693 soils. However, the foundations must be proportioned so that the critical stresses (see Figure 13.5) in the field are never exceeded. A factor of safety of about 2.5 to 3 should be used to calculate the allowable soil pressure, or sall r 5 scrr FS (13.8) where sall r 5 allowable soil pressure FS 5 factor of safety The differential and total settlements of these foundations should be similar to those of foundations designed for sandy soils. Continuous foundations may be safer than isolated foundations over collapsible soils in that they can effectively minimize differential settlement. Figure 13.6 shows a typical procedure for constructing continuous foundations. This procedure uses footing beams and longitudinal load-bearing beams. In the construction of heavy structures, such as grain elevators, over collapsible soils, settlements up to about 0.3 m are sometimes allowed (Peck, Hanson, and Thornburn, 1974). In this case, tilting of the foundation is not likely to occur, because there is no eccentric loading. The total expected settlement for such structures can be estimated from standard consolidation tests on specimens of field moisture content. Without eccentric loading, the foundations will exhibit uniform settlement over loessial deposits; however, if the soil is of residual or colluvial nature, settlement may not be uniform. The reason is the nonuniformity generally encountered in residual soils. Extreme caution must be used in building heavy structures over collapsible soils. If large settlements are expected, drilled-shaft and pile foundations should be considered. These types of foundation can transfer the load to a stronger load-bearing stratum. Load-bearing beams Figure 13.6 Continuous foundation with load-bearing beams (From Clemence, S. P., and Finbarr, A. O. (1981). “Design Considerations for Collapsible Soils,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 107, GT3 pp. 305–317. With permission from ASCE.) 694 Chapter 13: Foundations on Difficult Soils Foundation Design in Soils Susceptible to Wetting If the upper layer of soil is likely to get wet and collapse sometime after construction of the foundation, several design techniques to avoid failure of the foundation may be considered. They are as follow: Dynamic Compaction If the expected depth of wetting is about 1.5 to 2 m from the ground surface, the soil may be moistened and recompacted by heavy rollers. Spread footings and mats may be constructed over the compacted soil. An alternative to recompaction by heavy rollers is heavy tamping, which is sometimes referred to as dynamic compaction. (See Chapter 14.) Heavy tamping consists primarily of dropping a heavy weight repeatedly on the ground. The height of the drop can vary from 8 to 30 m. The stress waves generated by the dropping weight help in the densification of the soil. Lutenegger (1986) reported the use of dynamic compaction to stabilize a thick layer of friable loess before construction of a foundation in Russe, Bulgaria. During field exploration, the water table was not encountered to a depth of 10 m, and the natural moisture content was below the plastic limit. Initial density measurements made on undisturbed soil specimens indicated that the moisture content at saturation would exceed the liquid limit, a property usually encountered in collapsible loess. For dynamic compaction of the soil, the upper 1.7 m of crust material was excavated. A circular concrete weight of 133 kN was used as a hammer. At each grid point, compaction was achieved by dropping the hammer 7 to 12 times through a vertical distance of 2.5 m. Figure 13.7 shows the dry unit weight of the soil before and after compaction. The increase in dry unit weight of the soil shows that dynamic compaction can be used effectively to stabilize collapsible soil. Dry unit weght (kN/m3) 12 0 14 16 18 20 Before compaction 1 After compaction 2 Depth (m) 13.6 3 4 5 Figure 13.7 Dynamic compaction of a friable loess layer in Russe, Bulgaria (Adapted after Lutenegger, 1986) 13.7 General Nature of Expansive Soils 695 Chemical Stabilization If conditions are favorable, foundation trenches can be flooded with solutions of sodium silicate and calcium chloride to stabilize the soil chemically. The soil will then behave like a soft sandstone and resist collapse upon saturation. This method is successful only if the solutions can penetrate to the desired depth; thus, it is most applicable to fine sand deposits. Silicates are rather costly and are not generally used. However, in some parts of Denver, silicates have been used successfully. The injection of a sodium silicate solution to stabilize collapsible soil deposits has been used extensively in the former Soviet Union and Bulgaria (Houston and Houston, 1989). This process, which is used for dry collapsible soils and for wet collapsible soils that are likely to compress under the added weight of the structure to be built, consists of three steps: Step 1. Injection of carbon dioxide to remove any water that is present and for preliminary activation of the soil Step 2. Injection of sodium silicate grout Step 3. Injection of carbon dioxide to neutralize alkalies. Vibrofloation and Ponding When the soil layer is susceptible to wetting to a depth of about 10 m (<30 ft), several techniques may be used to cause collapse of the soil before construction of the foundation is begun. Two of these techniques are vibroflotation and ponding (also called flooding). Vibroflotation is used successfully in free-draining soil. (See Chapter 14.) Ponding—by constructing low dikes—is utilized at sites that have no impervious layers. However, even after saturation and collapse of the soil by ponding, some additional settlement of the soil may occur after construction of the foundation is begun. Additional settlement may also be caused by incomplete saturation of the soil at the time of construction. Ponding may be used successfully in the construction of earth dams. Extending Foundation Beyond Zone of Wetting If precollapsing the soil is not practical, foundations may be extended beyond the zone of possible wetting, although the technique may require drilled shafts and piles. The design of drilled shafts and piles must take into consideration the effect of negative skin friction resulting from the collapse of the soil structure and the associated settlement of the zone of subsequent wetting. In some cases, a rock-column type of foundation (vibroreplacement) may be considered. Rock columns are built with large boulders that penetrate the collapsible soil layer. They act as piles in transferring the load to a more stable soil layer. Expansive Soils 13.7 General Nature of Expansive Soils Many plastic clays swell considerably when water is added to them and then shrink with the loss of water. Foundations constructed on such clays are subjected to large uplifting forces caused by the swelling. These forces induce heaving, cracking, and the breakup of both building foundations and slab-on-grade members. Figure 13.8 shows 696 Chapter 13: Foundations on Difficult Soils Figure 13.8 Cracks in a wall due to heaving of an expansive clay (Courtesy of Anand Puppala, University of Texas at Arlington, Arlington, Texas) the cracks in a wall due to excessive heaving. Expansive clays cover large parts of the United States, South America, Africa, Australia, and India. In the United States, these clays are predominant in Texas, Oklahoma, and the upper Missouri Valley. In general, expansive clays have liquid limits and plasticity indices greater than about 40 and 15, respectively. As noted, the increase and decrease in moisture content causes clay to swell and shrink. Figure 13.9 shows shrinkage cracks on the ground surface of a clay weathered from the Eagle Ford shale formation in the Dallas-Fort Worth, Texas area. The depth in a soil to which periodic changes of moisture occur is usually referred to as the active zone (see Figure 13.10). The depth of the active zone varies, depending on the location of the site. Some typical active-zone depths in American cities are given in Table 13.3. In some clays and clay shales in the western United States, the depth of the active zone can be as much as 15 m. The active-zone depth can easily be determined by plotting the liquidity index against the depth of the soil profile over several seasons. Figure 13.11 shows such a plot for the Beaumont formation in the Houston area. 13.7 General Nature of Expansive Soils Figure 13.9 Shrinkage cracks on ground surface in a clay weathered from Eagle Ford shale formation in the Dallas-Fort Worth area (Courtesy of Thomas M. Petry, Missouri University of Science and Technology, Rolla, Missouri) Ground surface Moisture content Seasonal variation of moisture content Active zone (Depth z) um nt libri e Equi ure cont t mois Depth Figure 13.10 Definition of active zone 697 698 Chapter 13: Foundations on Difficult Soils Table 13.3 Typical Active-Zone Depths in Some U.S. Citiesa City Depth of active zone (m) Houston Dallas San Antonio Denver 1.5 to 3 2.1 to 4.6 3 to 9 3 to 4.6 a After O’Neill and Poormoayed (1980) (O’Neill, M. W., and Poormoayed, N.(1980). “Methodology for Foundations on Expansive Clays,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 106, No. GT12, pp. 1345–1367. With permission from ASCE.) Shrinkage cracks can extend deep into the active zone. Figure 13.12 shows interconnected shrinkage cracks extending from the ground surface into the active zone in an expansive clay. To study the magnitude of possible swell in a clay, simple laboratory oedometer tests can be conducted on undisturbed specimens. Two common tests are the unrestrained swell test and the swelling pressure test. They are described in the following sections. 1 Liquidity index 0 1 1 Depth (m) 2 Approximate depth of seasonal change, 1.67 m 3 Range over several seasons 4 5 6 Figure 13.11 Active zone in Houston area, Beaumont formation (O’Neill, M. W. and Poormoayed, N. (1980). “Methodology for Foundations on Expansive Clays,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 106, No. GT12, pp. 1345–1367. With permission from ASCE.) 13.8 Unrestrained Swell Test 699 Figure 13.12 Interconnected shrinkage cracks extended from the ground surface into the active zone (Courtesy of Thomas M. Petry, Missouri University of Science and Technology, Rolla, Missouri) 13.8 Unrestrained Swell Test In the unrestrained swell test, the specimen is placed in an oedometer under a small surcharge of about 6.9 kN>m2 (1 lb>in2 ). Water is then added to the specimen, and the expansion of the volume of the specimen (i.e., its height; the area of cross section is constant) is measured until equilibrium is reached. The percentage of free swell may than be expressed as the ratio sw(free) (%) 5 DH (100) H (13.9) where sw(free) 5 free swell, as a percentage DH 5 height of swell due to saturation H 5 original height of the specimen Vijayvergiya and Ghazzaly (1973) analyzed various soil test results obtained in this manner and prepared a correlation chart of the free swell, liquid limit, and natural moisture content, as shown in Figure 13.13. O’Neill and Poormoayed (1980) developed a relationship for calculating the free surface swell from this chart: DSF 5 0.0033Zsw(free) (13.10) 700 Chapter 13: Foundations on Difficult Soils 20 Percent swell, s(free) (%) 10 60 50 40 1 Liquid limit 70 0.1 0 40 10 20 30 Natural moisture content (%) 50 Figure 13.13 Relation between percentage of free swell, liquid limit, and natural moisture content (From Vijayvergiya, V. N. and Ghazzaly, O. I. (1973). “Prediction of Swelling Potential of Natural Clays,” Proceedings, Third International Research and Engineering Conference on Expansive Clays, pp. 227–234. With permission from ASCE.) where DSF 5 free surface swell Z 5 depth of active zone sw(free) 5 free swell, as a percentage 13.9 Swelling Pressure Test The swelling pressure can be determined from two different types of tests. They are • • Conventional consolidation test Constant volume test The two methods are briefly described here. Conventional Consolidation Test In this type of test, the specimen is placed in a oedometer under a small surcharge of about 6.9 kN/m2. Water is added to the specimen, allowing it to swell and reach an equilibrium position after some time. Subsequently, loads are added in convenient steps, and the specimen is consolidated. The plot of specimen deformation (␦) versus log sr is shown in Figure 13.14. The ␦ versus log sr plot crosses the horizontal line through the point of initial condition. The pressure corresponding to the point of intersection is the zero swell pressure, ssw r . 13.9 Swelling Pressure Test 701 Specimen deformation, d Swelling due to addition of water +ve 0 Consolidation test Initial condition –ve ssw Pressure, s Figure 13.14 Zero swell pressure from conventional consolidation test Constant Volume Test The constant volume test can be conducted by taking a specimen in a consolidation ring and applying a pressure equal to the effective overburden pressure, sor , plus the approximate anticipated surcharge caused by the foundation, ssr . Water is then added to the specimen. As the specimen starts to swell, pressure is applied in small increments to prevent swelling. Pressure is maintained until full swelling pressure is developed on the specimen, at which time the total pressure is ssw r 5 sor 1 ssr 1 s1r (13.11) where ssw r 5 total pressure applied to prevent swelling, or zero swell pressure s1r 5 additional pressure applied to prevent swelling after addition of water Figure 13.15 shows the variation of the percentage of swell with effective pressure during a swelling pressure test. (For more information on this type of test, see Sridharan et al., 1986.) Swell, sw (%) sw (1) Unloading 0 o s sw Effective pressure Figure 13.15 Swelling pressure test 702 Chapter 13: Foundations on Difficult Soils A ssw r of about 20 to 30 kN>m2 is considered to be low, and a ssw r of 1500 to 2000 kN>m2 is considered to be high. After zero swell pressure is attained, the soil specimen can be unloaded in steps to the level of the effective overburden pressure, sor . The unloading process will cause the specimen to swell. The equilibrium swell for each pressure level is also recorded. The variation of the swell, sw in percent, and the applied pressure on the specimen will be like that shown in Figure 13.15. The constant volume test can be used to determine the surface heave, DS, for a foundation (O’Neill and Poormoayed, 1980) as given by the formula n DS 5 a 3sw(1) (%)4 (Hi ) (0.01) (13.12) i51 where sw(1) (%) 5 swell, in percent, for layer i under a pressure of sor 1 ssr (see Figure 13.15) DHi 5 thickness of layer i ⬘ ) obtained from the It is important to point out that the zero swell pressure (sw conventional consolidation test and the constant volume test may not be the same. Table 13.4 summarizes some laboratory test results of Sridharan et al. (1986) to illustrate this point. It also was shown by Sridharan et al. (1986) that the zero swell pressure is a function of the dry unit weight of soil, but not of the initial moisture content (Figure 13.16). Table 13.4 Comparison of Zero Swell Pressure Obtained from Conventional Consolidation Tests and Constant Volume Tests—Summary of Test Results of Sridharan et al. (1986) ⬘ (kN/m2) sw Soil Liquid limit Plasticity index Initial void ratio, ei Consolidation test Constant volume test BC-1 BC-4 BC-5 BC-7 BC-8 80 98 96 96 94 44 57 65 66 62 0.893 1.002 0.742 0.572 0.656 294.3 382.6 500.3 1275.3 147.2 186.4 251.8 304.1 372.8 68.7 Example 13.1 A soil profile has an active zone of expansive soil of 1.83 m. The liquid limit and the average natural moisture content during the construction season are 50% and 20%, respectively. Determine the free surface swell. Zero swell pressure, ssw (kN/m2) 13.9 Swelling Pressure Test 703 500 Symbol 400 300 Moisture content 0 8.2 15.8 18.2 200 100 0 11 12 13 14 15 15.7 Dry unit weight (kN/m3) Figure 13.16 Plot of zero swell pressure with the dry unit weight of soil (After Sridharan et al., 1986. Copyright ASTM INTERNATIONAL. Reprinted with permission.) Solution From Figure 13.13 for LL 5 50% and w 5 20%, sw(free) 5 3%. From Eq. (13.10), DSF 5 0.0033Zsw(free) Hence, DSF 5 0.0033(1.83) (3) (12) 5 18.12 mm ■ Example 13.2 A soil profile’s active-zone depth is 3.5 m. If a foundation is to be placed 0.5 m below the ground surface, what would be the estimated total swell? The following data were obtained from laboratory tests: Depth (m) Swell under effective overburden and estimated foundation surcharge pressure, sw(1)(%) 0.5 1 2 3 2 1.5 0.75 0.25 Solution The values of sw(1) (%) are plotted with depth in Figure 13.17a. The area of this diagram will be the total swell. The trapezoidal rule yields DS 5 1 1 1 1 c (1) (0 1 0.5) 1 (1) (0.5 1 1.1) 1 (1) (1.1 1 2) d 100 2 2 2 5 0.026 m 5 26 mm 704 Chapter 13: Foundations on Difficult Soils 0 Ground surface 2 Swel1, sw(1) (%) 1 0.5 1.0 0.5 0 26 1.5 1.1 1.5 2.0 Total Swell, DS (mm) 1.1 m 1.5 0.75 2.5 2.5 0.5 3.0 0.25 3.5 3.5 Depth (m) Depth (m) (a) (b) Figure 13.17 (a) Variation of sw(1) with depth; (b) variation of ⌬S with depth ■ Example 13.3 In Example 13.2, if the allowable total swell is 10 mm, what would be the undercut necessary to reduce the total swell? Solution Using the procedure outlined in Example 13.2, we calculate the total swell at various depths below the foundation from Figure 13.17a as follows: Total swell, ⌬S (mm) Depth (m) 3.5 0 3 01 c 2.5 1.5 0.5 1 1 (0.5) (0.25) d 5 0.000625 m 5 0.625 mm 2 100 1 1 0.000625 1 c (0.5) (0.25 1 0.5) d 5 0.0025 m 5 2.5 mm 100 2 1 1 0.0025 1 c (1) (0.5 1 1.1) d 5 0.0105 m 5 10.5 mm 100 2 26 mm Plotted in Figure 13.17b, these total settlements show that a total swell of 10 mm corresponds to a depth of 1.6 m below the ground surface. Hence, the undercut below the foundation is 1.6 ⫺ 0.5 ⫽ 1.1 m. This soil should be excavated, replaced by nonswelling soil, and recompacted. ■ 13.10 Classification of Expansive Soil on the Basis of Index Tests Classification of Expansive Soil on the Basis of Index Tests Classification systems for expansive soils are based on the problems they create in the construction of foundations (potential swell). Most of the classifications contained in the literature are summarized in Figure 13.18 and Table 13.5. However, the classification system developed by the U.S. Army Waterways Experiment Station (Snethen et al., 1977) is the one most widely used in the United States. It has also been summarized by O’Neill and Poormoayed (1980); see Table 13.6. Sridharan (2005) proposed an index called the free swell ratio to predict the clay type, potential swell classification, and dominant clay minerals present 3 Very High 2 High 1 Medium Low 0 Swelling Potential 25% 5% 1.5% Plasticity index (%) Activity 100 80 n U 40 Li 0 Suction(pF) Very High 0 50 100 Percent of clay (2 ) in whole sample (c) 0.7 L 20 40 60 80 100 120 140 160 Liquid limit (%) (b) 7 Low A ine ) 0 I II III IV V 6 0 ( 0. 20 L- L 3( 20 100 High Medium e Extra high ) -8 LL 9 60 0 10 20 30 40 50 60 70 80 90 100 Percent clay sizes (finer than 0.002 mm) (a) 50 Very high High 120 Med 4 Swelling Low Nonplastic 5 Plasticity index of whole sample 13.10 705 5 Special case High Moderate Low Nonexpansive 4 I 3 V IV III II 2 1 0 10 20 30 40 50 Soil water content (d) 60 Figure 13.18 Commonly used criteria for determining swell potential (From Abduljauwad, S. N. and Al-Sulaimani, G. J. (1993). “Determination of Swell Potential of Al-Qatif Clay,” Geotechnical Testing Journal, American Society for Testing and Materials, vol. 16, No. 4, pp. 469–484. Copyright ASTM INTERNATIONAL. Reprinted with permission.) Table 13.5 Summary of Some Criteria for Identifying Swell Potential (From Abduljauwad, S. N. and Al-Sulaimani, G. J. (1993). “Determination of Swell Potential of Al-Qatif Clay,” Geotechnical Testing Journal, American Society for Testing and Materials, vol. 16, No. 4, pp. 469–484. Copyright ASTM INTERNATIONAL. Reprinted with permission.) Reference Criteria Remarks Holtz (1959) CC . 28, PI . 35, and SL , 11 (very high) 20 < CC < 31, 25 < PI < 41, and 7 < SL < 12 (high) 13 < CC < 23, 15 < PI < 28, and 10 < SL < 16 (medium) CC < 15, PI < 18, and SL > 15 (low) Based on CC, PI, and SL Seed et al. (1962) See Figure 13.18a Based on oedometer test using compacted specimen, percentage of clay ,2 mm, and activity Altmeyer (1955) LS , 5, SI . 12, and PS , 0.5 (noncritical) 5 < LS < 8, 10 < SL < 12, and 0.5 < PS < 1.5 (marginal) LS . 8, SL , 10, and PS . 1.5 (critical) Based on LS, SL, and PS Remolded sample (rd(max) and wopt) soaked under 6.9 kPa surcharge Dakshanamanthy See Figure 13.18b and Raman (1973) Based on plasticity chart Raman (1967) PI . 32 and SI . 40 (very high) 23 < PI < 32 and 30 < SI < 40 (high) 12 < PI < 23 and 15 < SI < 30 (medium) PI , 12 and SI , 15 (low) Based on PI and SI Sowers and Sowers (1970) SL , 10 and PI . 30 (high) 10 < SL < 12 and 15 < PI < 30 (moderate) SL . 12 and PI , 15 (low) Little swell will occur when wo results in LI of 0.25 Van Der Merwe (1964) See Figure 13.18c Based on PI, percentage of clay ,2 mm, and activity Uniform Building Code, 1968 EI . 130 (very high) and 91 < EI < 130 (high) 51 < EI < 90 (medium) and 21 < EI < 50 (low) 0 < EI < 20 (very low) Based on oedometer test on compacted specimen with degree of saturation close to 50% and surcharge of 6.9 kPa Snethen (1984) LL . 60, PI . 35, tnat . 4, and SP . 1.5 (high) 30 < LL < 60, 25 < PI < 35, 1.5 < tnat < 4, and 0.5 < SP < 1.5 (medium) LL , 30, PI , 25, tnat , 1.5, and SP , 0.5 (low) PS is representative for field condition and can be used without tnat , but accuracy will be reduced Chen (1988) PI $ 35 (very high) and 20 < PI < 55 (high) 10 < PI < 35 (medium) and PI < 15 (low) Based on PI McKeen (1992) Figure 13.18d Based on measurements of soil water content, suction, and change in volume on drying Vijayvergiya and Ghazzaly (1973) log SP 5 (1>12) (0.44 LL 2 wo 1 5.5) Empirical equations Nayak and Christensen (1974) SP 5 (0.00229 PI) (1.45C)>wo 1 6.38 Empirical equations Weston (1980) SP 5 0.00411(LL w ) 4.17q23.86w22.33 o Empirical equations 706 13.10 Classification of Expansive Soil on the Basis of Index Tests 707 Table 13.5 (continued) Note: C 5 clay, % CC 5 colloidal content, % EI 5 Expansion index 5 100 3 percent swell 3 fraction passing No. 4 sieve LI 5 liquidity index, % LL 5 liquid limit, % LL w 5 weighted liquid limit, % LS 5 linear shrinkage, % PI 5 plasticity index, % PS 5 probable swell, % q 5 surcharge SI 5 shrinkage index 5 LL 2 SL, % SL 5 shrinkage limit, % SP 5 swell potential, % wo 5 natural soil moisture wopt 5 optimum moisture content, % tnat 5 natural soil suction in tsf rd(max) 5 maximum dry density Table 13.6 Expansive Soil Classification Systema Liquid limit Plasticity index Potential swell (%) Potential swell classification Low ,50 ,25 ,0.5 50–60 25–35 0.5–1.5 Marginal High .60 .35 .1.5 Potential swell 5 vertical swell under a pressure equal to overburden pressure a Compiled from O’Neill and Poormoayed (1980) in a given soil. The free swell ratio can be determined by finding the equilibrium sediment volumes of 10 grams of an oven-dried specimen passing No. 40 U.S. sieve (0.425 mm opening) in distilled water (Vd) and in CCl4 or kerosene (VK). The free swell ratio (FSR) is defined as FSR 5 Vd VK (13.13) Table 13.7 gives the expansive soil classification based on free swell ratio. Also, Figure 13.19 shows the classification of soil based on the free swell ratio. Table 13.7 Expansive Soil Classification Based on Free Swell Ratio Free swell ratio < 1.0 1.0–1.5 1.5–2.0 2.0–4.0 . 4.0 Clay type Non-swelling Mixture of swelling and non-swelling Swelling Swelling Swelling Potential swell classification Dominant clay mineral Negligible Low Kaolinite Kaolinite and montmorillonite Montmorillonite Montmorillonite Montmorillonite Moderate High Very High 708 Chapter 13: Foundations on Difficult Soils 80 4 2 1 1 60 III C III B III A I Kaolinitic soils V d (cm3) 1.5 II (Kaolinitic + montmorillonitic) soils 1 III Montmorillonitic soils 40 A Moderately swelling 1 II B Highly swelling 1 C Very highly swelling 20 I 0 0 20 VK (cm3) 40 Figure 13.19 Classification based on free swell ratio (Adapted from Sridharan, 2005) 13.11 Foundation Considerations for Expansive Soils If a soil has a low swell potential, standard construction practices may be followed. However, if the soil possesses a marginal or high swell potential, precautions need to be taken, which may entail 1. Replacing the expansive soil under the foundation 2. Changing the nature of the expansive soil by compaction control, prewetting, installation of moisture barriers, or chemical stabilization 3. Strengthening the structures to withstand heave, constructing structures that are flexible enough to withstand the differential soil heave without failure, or constructing isolated deep foundations below the depth of the active zone One particular method may not be sufficient in all situations. Combining several techniques may be necessary, and local construction experience should always be considered. Following are some details regarding the commonly used techniques for dealing with expansive soils. 13.11 Foundation Considerations for Expansive Soils 709 Replacement of Expansive Soil When shallow, moderately expansive soils are present at the surface, they can be removed and replaced by less expansive soils and then compacted properly. Changing the Nature of Expansive Soil 1. Compaction: The heave of expansive soils decreases substantially when the soil is compacted to a lower unit weight on the high side of the optimum moisture content (possibly 3 to 4% above the optimum moisture content). Even under such conditions, a slab-on-ground type of construction should not be considered when the total probable heave is expected to be about 38 mm or more. 2. Prewetting: One technique for increasing the moisture content of the soil is ponding and hence achieving most of the heave before construction. However, this technique may be time consuming because the seepage of water through highly plastic clays is slow. After ponding, 4 to 5% of hydrated lime may be added to the top layer of the soil to make it less plastic and more workable (Gromko, 1974). 3. Installation of moisture barriers: The long-term effect of the differential heave can be reduced by controlling the moisture variation in the soil. This is achieved by providing vertical moisture barriers about 1.5 m deep around the perimeter of slabs for the slab-ongrade type of construction. These moisture barriers may be constructed in trenches filled with gravel, lean concrete, or impervious membranes. 4. Stabilization of soil: Chemical stabilization with the aid of lime and cement has often proved useful. A mix containing about 5% lime is sufficient in most cases. The effect of lime in stabilizing expansive soils, thereby reducing the shrinking and swelling characteristics, can be demonstrated with the aid of Figure 13.20. For this, expansive clay weathered from the Eagle Ford shale formation in the DallasFort Worth, Texas area was taken. Some of it was mixed with water to about its liquid limit. It was placed in two molds that were about 152 mm long and 12.7 mm ⫻ 12.7 mm in cross section. Figure 13.20a shows the shrinkage of the soil specimens in the mold in a dry condition. The same soil also was mixed with 6% lime (by dry weight) and then with a similar amount of water and placed in six similar molds. Figures 13.20b shows the shrinkage of the lime-stabilized specimens in a dry condition, which was practically negligible compared to that seen in Figure 13.20a. Lime or cement and water are mixed with the top layer of soil and compacted. The addition (a) Figure 13.20 Shrinkage of expansive clay (Eagle Ford soil) mixed with water to about its liquid limit in molds of 152 mm ⫻ 12.7 mm ⫻ 12.7 mm: (a) without addition of lime; 710 Chapter 13: Foundations on Difficult Soils Figure 13.20 (continued) (b) with addition of 6% lime by weight (Courtesy of Thomas M. Petry, Missouri University of Science and Technology, Rolla, Missouri) (b) of lime or cement will decrease the liquid limit, the plasticity index, and the swell characteristics of the soil. This type of stabilization work can be done to a depth of 1 to 1.5 m. Hydrated high-calcium lime and dolomite lime are generally used for lime stabilization. Another method of stabilization of expansive soil is the pressure injection of lime slurry or lime–fly-ash slurry into the soil, usually to a depth of 4 to 5 m or and occasionally deeper to cover the active zone. Further details of the pressure injection technique are presented in Chapter 14. Depending on the soil conditions at a site, single or multiple injections can be planned, as shown in Figure 13.21. Figure 13.22 shows the slurry pressure injection work for a building pad. The stakes that are marked are the planned injection points. Figure 13.23 shows lime–fly-ash stabilization by pressure injection of the bank of a canal that had experienced sloughs and slides. Plan Section Single injection Double injection Figure 13.21 Multiple lime slurry injection planning for a building pad 13.12 Construction on Expansive Soils 711 Figure 13.22 Pressure injection of lime slurry for a building pad (Courtesy of Hayward Baker Inc., Odenton, Maryland.) Figure 13.23 Slope stabilization of a canal bank by pressure injection of lime–fly-ash slurry (Courtesy of Hayward Baker Inc., Odenton, Maryland.) 13.12 Construction on Expansive Soils Care must be exercised in choosing the type of foundation to be used on expansive soils. Table 13.8 shows some recommended construction procedures based on the total predicted heave, DS, and the length-to-height ratio of the wall panels. For example, the table proposes the use of waffle slabs as an alternative in designing rigid buildings that are capable of tolerating movement. Figure 13.24 shows a schematic diagram of a waffle slab. In this type of construction, the ribs hold the structural load. The waffle voids allow the expansion of soil. 712 Chapter 13: Foundations on Difficult Soils Table 13.8 Construction Procedures for Expansive Clay Soilsa Total predicted heave (mm) L, H 5 1.25 L, H 5 2.5 0 to 6.35 6.35 to 12.7 12.7 12.7 to 50.8 12.7 to 50.8 50.8 to 101.6 Recommended construction Method Remarks No precaution Rigid building tolerating movement (steel reinforcement as necessary) Foundations: Pads Strip footings mat (waffle) Footings should be small and deep, consistent with the soilbearing capacity. Mats should resist bending. Floor slabs: Waffle Tile Slabs should be designed to resist bending and should be independent of grade beams. Walls: Walls on a mat should be as flexible as the mat. There should be no vertical rigid connections. Brickwork should be strengthened with tie bars or bands. Joints: Clear Flexible Contacts between structural units should be avoided, or flexible, waterproof material may be inserted in the joints. Building damping movement Walls: Walls or rectangular building Flexible units should heave as a unit. Unit construction Steel frame Foundations: Three point Cellular Jacks .50.8 .101.6 Building independent of movement Cellular foundations allow slight soil expansion to reduce swelling pressure. Adjustable jacks can be inconvenient to owners. Threepoint loading allows motion without duress. Foundation drilled Smallest-diameter and widely shaft: spaced shafts compatible with the Straight shaft load should be placed. Bell bottom Clearance should be allowed under grade beams. Suspended floor: Floor should be suspended on grade beams 305 to 460 mm above the soil. a Gromko, G. J., (1974). “Review of Expansive Soils,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 100, No. GT6, pp. 667–687. With permission from ASCE. 13.12 Construction on Expansive Soils Void 713 Void Figure 13.24 Waffle slab Dead load, D Grade beam Ground surface z Ds U Active zone, Z Drilled shafts with bells (a) Db (b) Figure 13.25 (a) Construction of drilled shafts with bells and grade beam; (b) definition of parameters in Eq. (13.14) Table 13.8 also suggests the use of a drilled shaft foundation with a suspended floor slab when structures are constructed independently of movement of the soil. Figure 13.25a shows a schematic diagram of such an arrangement. The bottom of the shafts should be placed below the active zone of the expansive soil. For the design of the shafts, the uplifting force, U, may be estimated (see Figure 13.25b) from the equation U 5 pDsZssw r tan fps r (13.14) where Ds 5 diameter of the shaft Z 5 depth of the active zone fps r 5 effective angle of plinth–soil friction ssw r 5 pressure for zero swell (see Figures 13.14 and 13.15; ssw r 5 sor 1 ssr 1 s1r ) In most cases, the value of fps r varies between 10 and 20°. An average value of the zero horizontal swell pressure must be determined in the laboratory. In the absence of laboratory results, ssw r tan fps r may be considered equal to the undrained shear strength of clay, cu , in the active zone. 714 Chapter 13: Foundations on Difficult Soils The belled portion of the drilled shaft will act as an anchor to resist the uplifting force. Ignoring the weight of the drilled shaft, we have Qnet 5 U 2 D (13.15) cuNc p ¢ ≤ (D2b 2 D2s ) FS 4 (13.16) where Qnet 5 net uplift load D 5 dead load Now, Qnet < where cu 5 undrained cohesion of the clay in which the bell is located Nc 5 bearing capacity factor FS 5 factor of safety Db 5 diameter of the bell of the drilled shaft Combining Eqs. (13.15) and (13.16) gives U2D5 cuNc p ¢ ≤ (D2b 2 D2s ) FS 4 (13.17) Conservatively, from Tables 3.3 and 3.4, Nc < Nc(strip)Fcs 5 Nc(strip) ¢1 1 NqB NcL ≤ < 5.14¢1 1 1 ≤ 5 6.14 5.14 A drilled-shaft design is examined in Example 13.4. Example 13.4 Figure 13.26 shows a drilled shaft with a bell. The depth of the active zone is 5 m. The zero swell pressure of the swelling clay (ssw r ) is 450 kN> m2. For the drilled shaft, the dead load (D) is 600 kN and the live load is 300 kN. Assume fps r 5 12°. a. Determine the diameter of the bell, Db. b. Check the bearing capacity of the drilled shaft assuming zero uplift force. Solution Part a: Determining the Bell Diameter, Db The uplift force, Eq. (13.14), is U 5 pDsZssw r tan fps r 13.12 Construction on Expansive Soils 715 Dead load live load 900 kN Active zone 5m 800 mm 2m cu 450 kN/m2 Db Figure 13.26 Drilled shaft in a swelling clay Given: Z 5 5 m and ssw r 5 450 kN>m2. Then U 5 p(0.8) (5) (450)tan 12° 5 1202 kN Assume the dead load and live load to be zero, and FS in Eq. (13.17) to be 1.25. So, from Eq. (13.17), U5 1202 5 cuNc p ¢ ≤ (D2b 2 D2s ) FS 4 (450) (6.14) p ¢ ≤ (D2b 2 0.82 ); 1.25 4 Db 5 1.15 m The factor of safety against uplift with the dead load also should be checked. A factor of safety of at least 2 is desirable. So, from Eq. (13.17) cuNc ¢ FS 5 5 p ≤ (D2b 2 D2s ) 4 U2D p (450) (6.14) ¢ ≤ (1.152 2 0.82 ) 4 1202 2 600 5 2.46 + 2 — OK Part b: Check for Bearing Capacity Assume that U 5 0. Then Dead load 1 live load 5 600 1 300 5 900 kN Downward load per unit area 5 900 p ¢ ≤ (D2b ) 4 5 900 p ¢ ≤ (1.152 ) 4 5 866.5 kN>m2 716 Chapter 13: Foundations on Difficult Soils Net bearing capacity of the soil under the bell 5 qu(net) 5 cuNc 5 (450) (6.14) 5 2763 kN>m2 Hence, the factor of safety against bearing capacity failure is FS 5 2763 5 3.19 + 3 — OK 866.5 ■ Sanitary Landfills 13.13 General Nature of Sanitary Landfills Sanitary landfills provide a way to dispose of refuse on land without endangering public health. Sanitary landfills are used in almost all countries, to varying degrees of success. The refuse disposed of in sanitary landfills may contain organic, wood, paper, and fibrous wastes, or demolition wastes such as bricks and stones. The refuse is dumped and compacted at frequent intervals and is then covered with a layer of soil, as shown in Figure 13.27. In the compacted state, the average unit weight of the refuse may vary between 5 and 10 kN>m3. A typical city in the United States, with a population of 1 million, generates about 3.8 3 106 m3 of compacted landfill material per year. As property values continue to increase in densely populated areas, constructing structures over sanitary landfills becomes more and more tempting. In some instances, a visual site inspection may not be enough to detect an old sanitary landfill. However, construction of foundations over sanitary landfills is generally problematic because of poisonous gases (e.g., methane), excessive settlement, and low inherent bearing capacity. Soil cover Excavation for soil cover Landfill Original ground surface Figure 13.27 Schematic diagram of a sanitary landfill in progress 13.14 Settlement of Sanitary Landfills 13.14 717 Settlement of Sanitary Landfills Sanitary landfills undergo large continuous settlements over a long time. Yen and Scanlon (1975) documented the settlement of several landfill sites in California. After completion of the landfill, the settlement rate (Figure 13.28) may be expressed as m5 DHf (m) Dt (month) (13.18) where m 5 settlement rate Hf 5 maximum height of the sanitary landfill On the basis of several field observations, Yen and Scanlon determined the following empirical correlations for the settlement rate: 3for fill heights ranging from 12 to 24 m4 m 5 a 2 b log t1 3for fill heights ranging from 24 to 30 m4 m 5 c 2 d log t1 3for fill heights larger than 30 m4 m 5 e 2 f log t1 In these equations, (13.19) (13.20) (13.21) m is in m>mo(ft>mo). t1 is the median fill age, in months In SI units, the values of a, b, c, d, e, and f given in Eqs. (13.19) through (13.21) are Item SI a b c d e f 0.0268 0.0116 0.038 0.0155 0.0435 0.0183 Height of sanitary landfill Hf Hf Hf 2 t t1 tc 2 tc t t tc tc t tc Time, t Figure 13.28 Settlement of sanitary landfills 718 Chapter 13: Foundations on Difficult Soils The median fill age may be defined from Figure 13.28 as t1 5 t 2 tc 2 (13.22) where t 5 time from the beginning of the landfill tc 5 time for completion of the landfill Equations (13.19), (13.20), and (13.21) are based on field data from landfills for which tc varied from 70 to 82 months. To get an idea of the approximate length of time required for a sanitary landfill to undergo complete settlement, consider Eq. (13.19). For a fill 12 m high and for tc 5 72 months, m 5 0.0268 2 0.0116 log t1 so log t1 5 0.0268 2 m 0.0116 If m 5 0 (zero settlement rate), log t1 5 2.31, or t1 < 200 months. Thus, settlement will continue for t1 2 tc>2 5 200 2 36 5 164 months (<14 years) after completion of the fill—a fairly long time. This calculation emphasizes the need to pay close attention to the settlement of foundations constructed on sanitary landfills. A comparison of Eqs. (13.19) through (13.21) for rates of settlement shows that the value of m increases with the height of the fill. However, for fill heights greater than about 30 m, the rate of settlement should not be much different from that obtained from Eq. (13.21). The reason is that the decomposition of organic matter close to the surface is mainly the result of an anaerobic environment. For deeper fills, the decomposition is slower. Hence, for fill heights greater than about 30 m, the rate of settlement does not exceed that for fills that are about 30 m in height. Sowers (1973) also proposed a formula for calculating the settlement of a sanitary landfill, namely, DHf 5 aHf 11e log ¢ ts ≤ tr (13.23) where Hf 5 height of the fill e 5 void ratio a 5 a coefficient for settlement tr, ts 5 times (see Figure 13.28) DHf 5 settlement between times tr and ts The coefficients a fall between a 5 0.09e (for conditions favorable to decomposition) (13.24) Problems 719 and a 5 0.03e (for conditions unfavorable to decomposition) (13.25) Equation (13.23) is similar to the equation for secondary consolidation settlement. Problems 13.1 For leossial soil, given Gs 5 2.69. Plot a graph of gd (kN>m3 ) versus liquid limit to identify the zone in which the soil is likely to collapse on saturation. If a soil has a liquid limit of 33, Gs 5 2.69, and gd 5 13.5 kN>m3, is collapse likely to occur? 13.2 A collapsible soil layer in the field has a thickness of 3 m. The average effective overburden pressure on the soil layer is 62 kN>m2. An undisturbed specimen of this soil was subjected to a double oedometer test. The preconsolidation pressure of the specimen as determined from the soaked specimen was 84 kN>m2 . Is the soil in the field normally consolidated or preconsolidated? 13.3 An expansive soil has an active-zone thickness of 8 m. The natural moisture content of the soil is 20% and its liquid limit is 50. Calculate the free surface swell of the expansive soil upon saturation. 13.4 An expansive soil profile has an active-zone thickness of 5.2 m. A shallow foundation is to be constructed at a depth of 1.2 m below the ground surface. Based on the swelling pressure test, the following are given. Depth from ground surface (m) 1.2 2.2 3.2 4.2 5.2 Swell under overburden and estimated foundation surcharge pressure, sw(1) (%) 3.0 2.0 1.2 0.55 0.0 Estimate the total possible swell under the foundation. 13.5 Refer to Problem 13.4. If the allowable total swell is 15 mm., what would be the necessary undercut? 13.6 Repeat Problem 13.4 with the following: active zone thickness 5 6 m, depth of shallow foundation 5 1.5 m. Depth from ground surface (m) 1.5 2.0 3.0 4.0 5.0 6.0 Swell under overburden and estimated foundation surcharge pressure, sw (1) (%) 5.5 3.1 1.5 0.75 0.4 0.0 720 Chapter 13: Foundations on Difficult Soils 13.7 Refer to Problem 13.6. If the allowable total swell is 30 mm, what would be the necessary undercut? 13.8 Refer to Figure 13.25b. For the drilled shaft with bell, given: Thickness of active zone, Z 5 9.15 m Dead load 5 1334 kN Live load 5 267 kN Diameter of the shaft, Ds 5 1.07 m Zero swell pressure for the clay in the active zone 5 574.6 kN>m2 Average angle of plinth-soil friction, fps r 5 15° Average undrained cohesion of the clay around the bell 5 144.6 kN>m2 Determine the diameter of the bell, Db. A factor of safety of 3 against uplift is required with the assumption that dead load plus live load is equal to zero. 13.9 Refer to Problem 13.8. If an additional requirement is that the factor of safety against uplift is at least 3 with the dead load on (live load 5 0), what should be the diameter of the bell? References ABDULJAUWAD, S. N., and AL-SULAIMANI, G. J. (1993). “Determination of Swell Potential of Al-Qatif Clay,” Geotechnical Testing Journal, American Society for Testing and Materials,Vol. 16, No. 4, pp. 469 – 484. ALTMEYER, W. T. (1955). “Discussion of Engineering Properties of Expansive Clays,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 81, No. SM2, pp. 17–19. BENITES, L. A. (1968). “Geotechnical Properties of the Soils Affected by Piping near the Benson Area, Cochise County, Arizona,” M. S. Thesis, University of Arizona, Tucson. CHEN, F. H. (1988). Foundations on Expansive Soils, Elsevier, Amsterdam. CLEMENCE, S. P., and FINBARR, A. O. (1981). “Design Considerations for Collapsible Soils,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 107, No. GT3, pp. 305 – 317. CLEVENGER, W. (1958). “Experience with Loess as Foundation Material,” Transactions, American Society of Civil Engineers, Vol. 123, pp. 151–170. DAKSHANAMANTHY, V., and RAMAN, V. (1973). “A Simple Method of Identifying an Expansive Soil,” Soils and Foundations, Vol. 13, No. 1, pp. 97–104. DENISOV, N. Y. (1951). The Engineering Properties of Loess and Loess Loams, Gosstroiizdat, Moscow. FEDA, J. (1964). “Colloidal Activity, Shrinking and Swelling of Some Clays,” Proceedings, Soil Mechanics Seminar, Loda, Illinois, pp. 531–546. GIBBS, H. J. (1961). Properties Which Divide Loose and Dense Uncemented Soils, Earth Laboratory Report EM-658, Bureau of Reclamation, U.S. Department of the Interior, Washington, DC. GROMKO, G. J. (1974). “Review of Expansive Soils,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 100, No. GT6, pp. 667– 687. HANDY, R. L. (1973). “Collapsible Loess in Iowa,” Proceedings, Soil Science Society of America, Vol. 37, pp. 281–284. HOLTZ, W. G. (1959). “Expansive Clays—Properties and Problems,” Journal of the Colorado School of Mines, Vol. 54, No. 4, pp. 89 –125. HOLTZ, W. G., and HILF, J. W. (1961). “Settlement of Soil Foundations Due to Saturation,” Proceedings, Fifth International Conference on Soil Mechanics and Foundation Engineering, Paris, Vol. 1, 1961, pp. 673– 679. References 721 HOUSTON, W. N., and HOUSTON, S. L. (1989). “State-of-the-Practice Mitigation Measures for Collapsible Soil Sites,” Proceedings, Foundation Engineering: Current Principles and Practices, American Society of Civil Engineers, Vol. 1, pp. 161–175. JENNINGS, J. E., and KNIGHT, K. (1975). “A Guide to Construction on or with Materials Exhibiting Additional Settlements Due to ‘Collapse’ of Grain Structure,” Proceedings, Sixth Regional Conference for Africa on Soil Mechanics and Foundation Engineering, Johannesburg, pp. 99–105. LUTENEGGER, A. J. (1986). “Dynamic Compaction in Friable Loess,” Journal of Geotechnical Engineering, American Society of Civil Engineers, Vol. 112, No. GT6, pp. 663–667. LUTENEGGER, A. J., and SABER, R. T. (1988). “Determination of Collapse Potential of Soils,” Geotechnical Testing Journal, American Society for Testing and Materials, Vol. 11. No. 3, pp. 173–178. MCKEEN, R. G. (1992). “A Model for Predicting Expansive Soil Behavior,” Proceedings, Seventh International Conference on Expansive Soils, Dallas, Vol. 1, pp. 1– 6. NAYAK, N. V., and CHRISTENSEN, R. W. (1974). “Swell Characteristics of Compacted Expansive Soils,” Clay and Clay Minerals, Vol. 19, pp. 251–261. O’NEILL, M. W., and POORMOAYED, N. (1980). “Methodology for Foundations on Expansive Clays,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 106, No. GT12, pp. 1345–1367. PECK, R. B., HANSON, W. E., and THORNBURN, T. B. (1974). Foundation Engineering, Wiley, New York. PRIKLONSKII, V. A. (1952). Gruntovedenic, Vtoraya Chast’, Gosgeolzdat, Moscow. RAMAN, V. (1967). “Identification of Expansive Soils from the Plasticity Index and the Shrinkage Index Data,” The Indian Engineer, Vol. 11, No. 1, pp. 17–22. SEED, H. B., WOODWARD, R. J., JR., and LUNDGREN, R. (1962). “Prediction of Swelling Potential for Compacted Clays,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 88, No. SM3, pp. 53 –87. SEMKIN, V. V., ERMOSHIN, V. M., and OKISHEV, N. D. (1986). “Chemical Stabilization of Loess Soils in Uzbekistan,” Soil Mechanics and Foundation Engineering (trans. from Russian), Vol. 23, No. 5, pp. 196–199. SNETHEN, D. R. (1984). “Evaluation of Expedient Methods for Identification and Classification of Potentially Expansive Soils,” Proceedings, Fifth International Conference on Expansive Soils, Adelaide, pp. 22–26. SNETHEN, D. R., JOHNSON, L. D., and PATRICK, D. M. (1977). An Evaluation of Expedient Methodology for Identification of Potentially Expansive Soils, Report No. FHWA-RD-77-94, U.S. Army Engineers Waterways Experiment Station, Vicksburg, MS. SOWERS, G. F. (1973). “Settlement of Waste Disposal Fills,” Proceedings, Eighth International Conference on Soil Mechanics and Foundation Engineering, Moscow, pp. 207–210. SOWERS, G. B., and SOWERS, G. F. (1970). Introductory Soil Mechanics and Foundations, 3d ed. Macmillan, New York. SRIDHARAN, A. (2005). “On Swelling Behaviour of Clays,” Proceedings, International Conference on Problematic Soils, North Cyprus, Vol. 2, pp. 499–516. SRIDHARAN, A., RAO, A. S., and SIVAPULLAIAH, P. V. (1986), “Swelling Pressure of Clays,” Geotechnical Testing Journal, American Society for Testing and Materials, Vol. 9, No. 1, pp. 24–33. Uniform Building Code (1968). UBC Standard No. 29-2. VAN DER MERWE, D. H. (1964), “The Prediction of Heave from the Plasticity Index and Percentage Clay Fraction of Soils,” Civil Engineer in South Africa, Vol. 6, No. 6, pp. 103–106. VIJAYVERGIYA, V. N., and GHAZZALY, O. I. (1973). “Prediction of Swelling Potential of Natural Clays,” Proceedings, Third International Research and Engineering Conference on Expansive Clays, pp. 227–234. WESTON, D. J. (1980). “Expansive Roadbed Treatment for Southern Africa,” Proceedings, Fourth International Conference on Expansive Soils, Vol. 1, pp. 339–360. YEN, B. C., and SCANLON, B. (1975). “Sanitary Landfill Settlement Rates,” Journal of the Geotechnical Engineering Division, American Society of Civil Engineers, Vol. 101, No. GT5, pp. 475–487. 14 14.1 Soil Improvement and Ground Modification Introduction The soil at a construction site may not always be totally suitable for supporting structures such as buildings, bridges, highways, and dams. For example, in granular soil deposits, the in situ soil may be very loose and indicate a large elastic settlement. In such a case, the soil needs to be densified to increase its unit weight and thus its shear strength. Sometimes the top layers of soil are undesirable and must be removed and replaced with better soil on which the structural foundation can be built. The soil used as fill should be well compacted to sustain the desired structural load. Compacted fills may also be required in low-lying areas to raise the ground elevation for construction of the foundation. Soft saturated clay layers are often encountered at shallow depths below foundations. Depending on the structural load and the depth of the layers, unusually large consolidation settlement may occur. Special soil-improvement techniques are required to minimize settlement. In Chapter 13, we mentioned that the properties of expansive soils could be altered substantially by adding stabilizing agents such as lime. Improving in situ soils by using additives is usually referred to as stabilization. Various techniques are used to 1. Reduce the settlement of structures 2. Improve the shear strength of soil and thus increase the bearing capacity of shallow foundations 3. Increase the factor of safety against possible slope failure of embankments and earth dams 4. Reduce the shrinkage and swelling of soils This chapter discusses some of the general principles of soil improvement, such as compaction, vibroflotation, precompression, sand drains, wick drains, stabilization by admixtures, jet grouting, and deep mixing, as well as the use of stone columns and sand compaction piles in weak clay to construct foundations. 722 14.2 General Principles of Compaction 14.2 723 General Principles of Compaction If a small amount of water is added to a soil that is then compacted, the soil will have a certain unit weight. If the moisture content of the same soil is gradually increased and the energy of compaction is the same, the dry unit weight of the soil will gradually increase. The reason is that water acts as a lubricant between the soil particles, and under compaction it helps rearrange the solid particles into a denser state. The increase in dry unit weight with increase of moisture content for a soil will reach a limiting value beyond which the further addition of water to the soil will result in a reduction in dry unit weight. The moisture content at which the maximum dry unit weight is obtained is referred to as the optimum moisture content. The standard laboratory tests used to evaluate maximum dry unit weights and optimum moisture contents for various soils are • The Standard Proctor test (ASTM designation D-698) • The Modified Proctor test (ASTM designation D-1557) The soil is compacted in a mold in several layers by a hammer. The moisture content of the soil, w, is changed, and the dry unit weight, gd , of compaction for each test is determined. The maximum dry unit weight of compaction and the corresponding optimum moisture content are determined by plotting a graph of gd against w (%). The standard specifications for the two types of Proctor test are given in Tables 14.1 and 14.2. Table 14.1 Specifications for Standard Proctor Test (Based on ASTM Designation 698) Item Method A Method B Method C Diameter of mold Volume of mold Mass of hammer Height of hammer drop Number of hammer blows per layer of soil Number of layers of compaction Energy of compaction Soil to be used 101.6 mm 944 cm3 2.5 kg 304.8 mm 25 101.6 mm 944 cm3 2.5 kg 304.8 mm 25 152.4 mm 2124 cm3 2.5 kg 304.8 mm 56 3 3 3 600 kN # m>m3 Portion passing No. 4 4.57-mm sieve. May be used if 20% or less by weight of material is retained on No. 4 sieve. 600 kN # m>m3 Portion passing 9.5-mm sieve. May be used if soil retained on No. 4 sieve is more than 20% and 20% or less by weight is retained on 9.5-mm (38 -in.) sieve. 600 kN # m>m3 Portion passing 19.0-mm sieve. May be used if more than 20% by weight of material is retained on 9.5 mm sieve and less than 30% by weight is retained on 19.00-mm sieve. 724 Chapter 14: Soil Improvement and Ground Modification Table 14.2 Specifications for Modified Proctor Test (Based on ASTM Designation 1557) Item Method A Method B Method C Diameter of mold Volume of mold Mass of hammer Height of hammer drop Number of hammer blows per layer of soil Number of layers of compaction Energy of compaction Soil to be used 101.6 mm 944 cm3 4.54 kg 101.6 mm 944 cm3 4.54 kg 152.4 mm 2124 cm3 4.54 kg 457.2 mm 457.2 mm 457.2 mm 25 25 56 5 5 5 2700 kN # m>m3 2700 kN # m>m3 2700 kN # m>m3 Portion passing No. 4 (4.57-mm) sieve. May be used if 20% or less by weight of material is retained on No. 4 sieve. Portion passing 9.5-mm sieve. May be used if soil retained on No. 4 sieve is more than 20% and 20% or less by weight is retained on 9.5-mm sieve. Portion passing 19.0-mm (34 -in.) sieve. May be used if more than 20% by weight of material is retained on 9.5-mm sieve and less than 30% by weight is retained on 19-mm sieve. Figure 14.1 shows a plot of gd against w (%) for a clayey silt obtained from standard and modified Proctor tests (method A). The following conclusions may be drawn: 1. The maximum dry unit weight and the optimum moisture content depend on the degree of compaction. 2. The higher the energy of compaction, the higher is the maximum dry unit weight. 3. The higher the energy of compaction, the lower is the optimum moisture content. 4. No portion of the compaction curve can lie to the right of the zero-air-void line. The zero-air-void dry unit weight, gzav , at a given moisture content is the theoretical maximum value of gd , which means that all the void spaces of the compacted soil are filled with water, or gzav 5 where gw 5 unit weight of water Gs 5 specific gravity of the soil solids w 5 moisture content of the soil gw 1 1w Gs (14.1) 14.2 General Principles of Compaction 725 24 Dry unit weight, ␥d (kN/m3) 22 Zero-air-void curve (Gs 2.7) 20 18 16 14 Standard Proctor test 12 Modified Proctor test 10 0 5 10 15 20 Moisture content, w (%) 25 Figure 14.1 Standard and modified Proctor compaction curves for a clayey silt (method A) 5. The maximum dry unit weight of compaction and the corresponding optimum moisture content will vary from soil to soil. Using the results of laboratory compaction (gd versus w), specifications may be written for the compaction of a given soil in the field. In most cases, the contractor is required to achieve a relative compaction of 90% or more on the basis of a specific laboratory test (either the standard or the modified Proctor compaction test). The relative compaction is defined as RC 5 gd(field) (14.2) gd(max) Chapter 1 introduced the concept of relative density (for the compaction of granular soils), defined as Dr 5 B gd 2 gd(min) gd(max) 2 gd(min) R gd(max) gd 726 Chapter 14: Soil Improvement and Ground Modification where gd 5 dry unit weight of compaction in the field gd(max) 5 maximum dry unit weight of compaction as determined in the laboratory gd(min) 5 minimum dry unit weight of compaction as determined in the laboratory For granular soils in the field, the degree of compaction obtained is often measured in terms of relative density. Comparing the expressions for relative density and relative compaction reveals that RC 5 where A 5 gd(min) gd(max) A 1 2 Dr (1 2 A) (14.3) . Omar, et al. (2003) recently presented the results of modified Proctor compaction tests on 311 soil samples. Of these samples, 45 were gravelly soil (GP, GP-GM, GW, GW-GM, and GM), 264 were sandy soil (SP, SP-SM, SW-SM, SW, SC-SM, SC, and SM), and two were clay with low plasticity (CL). All compaction tests were conducted using ASTM 1557 method C to avoid over-size correction. Based on the tests, the following correlations were developed. rd(max ) (kg>m3 ) 5 34,804,574GS 2 195.55(LL) 2 1 156,971(R[ 4) 0.5 2 9,527,8304 0.5 24 2 (14.4) 25 ln(wopt ) 5 1.195 3 10 (LL) 2 1.964Gs 2 6.617 3 10 (R[ 4) 1 7.651 (14.5) where d(max) ⫽ maximum dry density wopt ⫽ optimum moisture content Gs ⫽ specific gravity of soil solids LL ⫽ liquid limit, in percent R # 4 ⫽ percent retained on No. 4 sieve It needs to be pointed out that Eqs. (14.4 and 14.5) contain the term for liquid limit. This is because the soils that were considered included silty and clayey sands. Osman et al. (2008) analyzed a number of laboratory compaction-test results on finegrained (cohesive) soil. Based on this study, the following correlations were developed: wopt 5 (1.99 2 0.165 ln E) (PI) (14.6) gd(max) 5 L 2 Mwopt (14.7) and 14.3 Field Compaction 727 where L ⫽ 14.34 ⫹ 1.195 ln E M ⫽ ⫺0.19 ⫹ 0.073 ln E (14.8) (14.9) wopt ⫽ optimum moisture content (%) PI ⫽ plasticity index (%) ␥d(max) ⫽ maximum dry unit weight (kN/m3) E ⫽ compaction energy (kN-m/m3) 14.3 Field Compaction Ordinary compaction in the field is done by rollers. Of the several types of roller used, the most common are 1. 2. 3. 4. Smooth-wheel rollers (or smooth drum rollers) Pneumatic rubber-tired rollers Sheepsfoot rollers Vibratory rollers Figure 14.2 shows a smooth-wheel roller that can also create vertical vibration during compaction. Smooth-wheel rollers are suitable for proof-rolling subgrades and for finishing the construction of fills with sandy or clayey soils. They provide 100% coverage under the wheels, and the contact pressure can be as high as Figure 14.2 Vibratory smooth-wheel rollers (Courtesy of Tampo Manufacturing Co., Inc., San Antonio, Texas) 728 Chapter 14: Soil Improvement and Ground Modification Figure 14.3 Pneumatic rubber-tired roller (Courtesy of Tampo Manufacturing Co., Inc., San Antonio, Texas) 300 to 400 kN>m2. However, they do not produce a uniform unit weight of compaction when used on thick layers. Pneumatic rubber-tired rollers (Figure 14.3) are better in many respects than smooth-wheel rollers. Pneumatic rollers, which may weigh as much as 2000 kN, consist of a heavily loaded wagon with several rows of tires. The tires are closely spaced— four to six in a row. The contact pressure under the tires may range up to 600 to 700 kN>m2, and they give about 70 to 80% coverage. Pneumatic rollers, which can be used for sandy and clayey soil compaction, produce a combination of pressure and kneading action. Sheepsfoot rollers (Figure 14.4) consist basically of drums with large numbers of projections. The area of each of the projections may be 25 to 90 cm2. These rollers are most effective in compacting cohesive soils. The contact pressure under the projections may range from 1500 to 7500 kN>m2. During compaction in the field, the initial passes compact the lower portion of a lift. Later, the middle and top of the lift are compacted. Vibratory rollers are efficient in compacting granular soils. Vibrators can be attached to smooth-wheel, pneumatic rubber-tired or sheepsfoot rollers to send vibrations into the soil being compacted. Figures 14.2 and 14.4 show vibratory smooth-wheel rollers and a vibratory sheepsfoot roller, respectively. In general, compaction in the field depends on several factors, such as the type of compactor, type of soil, moisture content, lift thickness, towing speed of the compactor, and number of passes the roller makes. Figure 14.5 shows the variation of the unit weight of compaction with depth for a poorly graded dune sand compacted by a vibratory drum roller. Vibration was 14.3 Field Compaction 729 Figure 14.4 Vibratory sheepsfoot roller (Courtesy of Tampo Manufacturing Co., Inc., San Antonio, Texas) 16 Dry unit weight (kN/m3) 16.5 17 0 Depth (m) 0.5 1.0 2 5 15 1.5 45 Number of roller passes 2.0 Figure 14.5 Vibratory compaction of a sand: Variation of dry unit weight with depth and number of roller passes; lift thickness 5 2.44 m (After D’Appolonia et al., 1969) (After D’Appolonia, D. J., Whitman, R. V. and D’Appolonia, E. (1969). “Sand Compaction with Vibratory Rollers,” Journal of the Soil Mechanics and Foundations Division, American Society of Civil Engineers, Vol. 95, N. SM1, pp. 263–284. With permission from ASCE.) 730 Chapter 14: Soil Improvement and Ground Modification produced by mounting an eccentric weight on a single rotating shaft within the drum cylinder. The weight of the roller used for this compaction was 55.7 kN, and the drum diameter was 1.19 m. The lifts were kept at 2.44 m. Note that, at any depth, the dry unit weight of compaction increases with the number of passes the roller makes. However, the rate of increase in unit weight gradually decreases after about 15 passes. Note also the variation of dry unit weight with depth by the number of roller passes. The dry unit weight and hence the relative density, Dr , reach maximum values at a depth of about 0.5 m and then gradually decrease as the depth increases. The reason is the lack of confining pressure toward the surface. Once the relation between depth and relative density (or dry unit weight) for a soil for a given number of passes is determined, for satisfactory compaction based on a given specification, the approximate thickness of each lift can be easily estimated. Hand-held vibrating plates can be used for effective compaction of granular soils over a limited area. Vibrating plates are also gang-mounted on machines. These can be used in less restricted areas. 14.4 Compaction Control for Clay Hydraulic Barriers Compacted clays are commonly used as hydraulic barriers in cores of earth dams, liners and covers of landfills, and liners of surface impoundments. Since the primary purpose of a barrier is to minimize flow, the hydraulic conductivity, k, is the controlling factor. In many cases, it is desired that the hydraulic conductivity be less than 1027cm>s. This can be achieved by controlling the minimum degree of saturation during compaction, a relation that can be explained by referring to the compaction characteristics of three soils described in Table 14.3 (Othman and Luettich, 1994). Figures 14.6, 14.7, and 14.8 show the standard and modified Proctor test results and the hydraulic conductivities of compacted specimens. Note that the solid symbols represent specimens with hydraulic conductivities of 1027cm>s or less. As can be seen from these figures, the data points plot generally parallel to the line of full saturation. Figure 14.9 shows the effect of the degree of saturation during compaction on the hydraulic conductivity of the three soils. It is evident from the figure that, if it is desired that the maximum hydraulic conductivity be 1027cm>s, then all soils should be compacted at a minimum degree of saturation of 88%. Table 14.3 Characteristics of Soils Reported in Figures 14.6, 14.7, and 14.8 Soil Classification Liquid limit Wisconsin A Wisconsin B Wisconsin C CL CL CH 34 42 84 Plasticity index Percent finer than No. 200 sieve (0.075 mm) 16 19 60 85 99 71 14.4 Compaction Control for Clay Hydraulic Barriers 731 Hydraulic conductivity (cm/s) 10–5 Standard Proctor Modified Proctor 10–6 10–7 10–8 10–9 10 12 14 16 18 Moisture content (%) (a) 20 22 24 Dry unit weight (kN/m3) 19 18 Sa tur ati on 17 10 0% 90 % 16 80 % Solid symbols represent specimens with hydraulic conductivity equal to or less than 1 10–7 cm/s 15 10 12 14 16 18 Moisture content (%) (b) 20 22 24 Figure 14.6 Standard and Modified Proctor test results and hydraulic conductivity of Wisconsin A soil (After Othman and Luettich, 1994) (From Othman, M. A., and S. M. Luettich. Compaction Control Criteria for Clay Hydraulic Barriers. In Transportation Research Record 1462, Transportation Research Board, National Research Council, Washington, D.C., 1994, Figures 4 and 5, p. 32, and Figures 6 and 7, p. 33. Reproduced with permission of the Transportation Research Board.) In field compaction at a given site, soils of various composition may be encountered. Small changes in the content of fines will change the magnitude of hydraulic conductivity. Hence, considering the various soils likely to be encountered at a given site the procedure just described aids in developing a minimum-degree-of-saturation criterion for compaction to construct hydraulic barriers. 732 Chapter 14: Soil Improvement and Ground Modification 10–5 Hydraulic conductivity (cm/s) Solid symbols represent specimens with hydraulic conductivity equal to or less than 1 10–7 cm/s 10–6 10–7 10–8 10–9 8 12 16 Moisture content (%) (a) 20 24 19 Figure 14.7 Standard and Modified Proctor test results and hydraulic conductivity of Wisconsin B soil (After Othman and Luettich, 1994) (From Othman, M. A., and S. M. Luettich. Compaction Control Criteria for Clay Hydraulic Barriers. In Transportation Research Record 1462, Transportation Research Board, National Research Council, Washington, D.C., 1994, Figures 4 and 5, p. 32, and Figures 6 and 7, p. 33. Reproduced with permission of the Transportation Research Board.) 14.5 Dry unit weight (kN/m3) Standard Proctor Modified Proctor 18 Sa 17 tur ati o n 10 0% 16 90 % 80 % 15 8 12 16 Moisture content (%) (b) 20 24 Vibroflotation Vibroflotation is a technique developed in Germany in the 1930s for in situ densification of thick layers of loose granular soil deposits. Vibroflotation was first used in the United States about 10 years later. The process involves the use of a vibroflot (called the vibrating unit), as shown in Figure 14.10. The device is about 2 m in length. This vibrating unit has an eccentric weight inside it and can develop a centrifugal force. The weight enables the unit to vibrate horizontally. Openings at the bottom and top of the unit are for water jets. The vibrating unit is attached to a follow-up pipe. The figure shows the vibroflotation equipment necessary for compaction in the field. 14.5 Vibroflotation 733 Hydraulic conductivity (cm/s) 10–5 Solid symbols represent specimens with hydraulic conductivity equal to or less than 1 10–7 cm/s 10–6 10–7 10–8 10–9 15 20 25 Moisture content (%) (a) 30 35 16.51 Standard Proctor Modified Proctor Figure 14.8 Standard and Modified Proctor test results and hydraulic conductivity of Wisconsin C soil (After Othman and Luettich, 1994) (From Othman, M. A., and S. M. Luettich. Compaction Control Criteria for Clay Hydraulic Barriers. In Transportation Research Record 1462, Transportation Research Board, National Research Council, Washington, D.C., 1994, Figures 4 and 5, p. 32, and Figures 6 and 7, p. 33. Reproduced with permission of the Transportation Research Board.) Dry unit weight (kN/m3) 16.0 Sa tu 15.0 ra ti on % % 15 20 25 Moisture content (%) (b) 10 0% 90 80 14.0 30 35 The entire compaction process can be divided into four steps (see Figure 14.11): Step 1. The jet at the bottom of the vibroflot is turned on, and the vibroflot is lowered into the ground. Step 2. The water jet creates a quick condition in the soil, which allows the vibrating unit to sink. Step 3. Granular material is poured into the top of the hole. The water from the lower jet is transferred to the jet at the top of the vibrating unit. This water carries the granular material down the hole. Step 4. The vibrating unit is gradually raised in about 0.3-m lifts and is held vibrating for about 30 seconds at a time. This process compacts the soil to the desired unit weight. 734 Chapter 14: Soil Improvement and Ground Modification Hydraulic conductivity (cm/s) 10–5 10–6 10–7 Soil A, Standard Proctor Soil A, Modified Proctor Soil B, Standard Proctor Soil B, Modified Proctor Soil C, Standard Proctor Soil C, Modified Proctor 10–8 10–9 40 50 60 70 80 Degree of saturation (%) 90 100 Figure 14.9 Effect of degree of saturation on hydraulic conductivity of Wisconsin A, B, and C soils (After Othman and Luettich, 1994) (From Othman, M. A., and S. M. Luettich. Compaction Control Criteria for Clay Hydraulic Barriers. In Transportation Research Record 1462, Transportation Research Board, National Research Council, Washington, D.C., 1994, Figures 4 and 5, p. 32, and Figures 6 and 7, p. 33. Reproduced with permission of the Transportation Research Board.) Table 14.4 gives the details of various types of vibroflot unit used in the United States. The 23 kW electric units have been used since the latter part of the 1940s. The 100-HP units were introduced in the early 1970s. The zone of compaction around a single probe will vary according to the type of vibroflot used. The cylindrical zone of compaction will have a radius of about 2 m for a 23 kW unit and about 3 m for a 75 kW unit. Compaction by vibroflotation involves various probe spacings, depending on the zone of compaction. (See Figure 14.12.) Mitchell (1970) and Brown (1977) reported several successful cases of foundation design that used vibroflotation. The success of densification of in situ soil depends on several factors, the most important of which are the grain-size distribution of the soil and the nature of the backfill used to fill the holes during the withdrawal period of the vibroflot. The range of the grain-size distribution of in situ soil marked Zone 1 in Figure 14.13 is most suitable for compaction by vibroflotation. Soils that contain excessive amounts of fine sand and silt-size particles are difficult to compact; for such soils, considerable effort is needed to reach the proper relative de